GB Patent 2395137 - A user selectable betting draw apparatus

Description

"A Gaming Apparatus" THIS INVENTION relates to a gaming apparatus, and

particularly concerns a gaming apparatus which allows a user to define parameters of a game upon which to place a bet.

One aspect of the present invention provides a gaming apparatus comprising: a pool size selector, allowing a user to determine a number of items of a first type and a number of items of a second type that are to be placed into a pool; a draw size selector, allowing the user to specify a number of items to be drawn from the pool; a bet selector, allowing the user to place a bet on an outcome of the drawing of the specified number of items from the pool; a drawer, operable to draw the specified number of items from the pool; and an outcome determiner, operable to establish whether the user has won the bet.

Advantageously, the items comprise physical objects.

Preferably, the objects of the first type are of a first colour, and the objects of the second type are of a second colour.

Conveniently, the objects comprise balls.

Alternatively, the items comprise virtual objects.

Advantageously, the apparatus further comprises a display upon which a representation of physical objects may be displayed, each of the represented physical objects corresponding to one of the virtual objects.

Preferably, the apparatus further comprises a calculator to calculate the odds to be offered for the bet placed by the user.

Another aspect of the present invention provides a method of manufacturing a game card, the method comprising the steps of: determining a number of items of a first type and a number of items of a second type that are to be placed into a pool; specifying a number of items to be drawn from the pool; printing details of the numbers of the of the first and second types of items placed in the pool and the number of items to be drawn from the pool on the card; drawing the specified number of items from the pool; and printing a representation of the drawn items on the card.

Conveniently, the method further comprises the step of applying a removable covering over the printed representation of the drawn items.

Advantageously, the method further comprises the step of printing odds relating to several possible outcomes of the draw on the card.

A further aspect of the present invention provides a method of producing a batch of game cards, comprising the steps of: manufacturing a batch of game cards by the method of any one of Claims 8 to 10; and reviewing the printed

details on the game cards to determine a measure of profitability of the batch as a whole.

Another aspect of the present invention provides a computer program comprising computer program code means adapted to: determine a number of items of a first type and a number of items of a second type that are to be placed into a pool; allow a user to specify a number of items to be drawn from the pool; allow the user to place a bet on an outcome of the drawing of the specified number of items from the pool; draw the specified number of items from the pool; and establish whether the user has won the bet, when said program is run on a computer.

A further aspect of the present invention provides a computer program comprising computer program code means adapted to: perform the above steps a plurality of times to generate a batch of results; and review the results to determine a measure of profitability of the batch as a whole, when said program is run on a computer.

Another aspect of the present invention provides a computer program comprising computer program code means adapted to: determine a number of items of a first type and a number of items of a second type that are to be placed into a pool; specify a number of items to be drawn from the pool; print details of the numbers of the of the first and second types of items placed in the pool and the number of items to be drawn from the pool on the card; draw the specified number of items from the pool; and print a representation of the drawn items on the card, when said program is run on a computer.

A further aspect of the present invention provides a computer program according to the above, embodied on a computer readable medium.

In order that the present invention may be more readily understood, embodiments will now be described, by way of example, with reference to the accompanying drawings, in which: Figure 1 shows a first gaming apparatus embodying the present invention; and Figure 2 shows a second gaming apparatus embodying the present invention. Referring firstly to Figure 1, a first gaming apparatus 1 embodying the present invention is shown. The first gaming apparatus 1 contains a chamber 2, and is preferably provided with a transparent chamber window 3 positioned so to allow a user of the apparatus 1 to see into the chamber 2.

The first gaming apparatus 1 is equipped with two reservoirs (not shown), a first reservoir containing a supply of red balls and a second reservoir containing a supply of black balls. While, in the present example, red balls and black balls are employed, any sets of items which may be distinguished from one another may be used. These items may be of any shape, and could, for instance, be numbered or otherwise marked rather than being provided in different colours. Of course, the invention is not limited to the provision of red and black objects.

The first gaming apparatus 1 is provided with appropriate tubes or other means operable to transfer a selected and precise number of red balls and black balls from the respective first and second reservoirs into the chamber. In preferred embodiments of the present invention, mixing means (not shown) may be provided within the chamber to mix or otherwise agitate the red and

black balls that have been transferred into the chamber 2, thereby ensuring a random arrangement of these balls.

The first gaming apparatus 1 is further provided with appropriate tubes or other means to draw a selected number of balls from the chamber. Each ball should be selected from the chamber 2 entirely at random and the provision of mixing means within the chamber 2 as described above is of assistance in this respect. A viewing window 4 is provided in the first gaming apparatus 1, and balls drawn from the chamber 2 are transferred to a location where they may be viewed through the viewing window 4. Preferably, balls selected from the chamber 2 come to rest in a tube or similar elongate chamber, in the order in which they are drawn from the chamber 2, and the viewing window 4 allows a user to view the balls lined up in the elongate chamber, and hence see whether each ball drawn from the chamber 2 is red or black.

The first gaming apparatus 1 is preferably further provided with returning means, which return balls that were drawn from the chamber 2, as well as balls that were not drawn therefrom, to the respective reservoirs.

Sorting means are employed to differentiate between the black and red balls and ensure that the balls are returned to the appropriate reservoirs, and this may be achieved by, for instance, a CCD camera operable to detect the colour of each ball as it passes, and a moveable surface activated in response to the detected colour to direct a returning ball along an appropriate path.

The front panel of the first gaming apparatus 1 comprises a pool size selector 5a, 5b, a draw size selector 6 and a bet selector 7, the functions of which will be described in greater detail below.

In use of the first gaming apparatus 1, a user is required to select the number of black balls and number of red balls that are to be placed into the chamber 2 prior to the start of the game. This is done using the pool size selector 5a, 5b, which allows the user to select separately the number of black balls and the number of red balls that are to be placed in the chamber 2. In a preferred embodiment, a pair of numerical keypads are provided to allow the user to key in the desired numbers. Preferably, immediately following the user's selection of a number of black and red balls, the appropriate number of balls are selected from the respective reservoirs and introduced into the chamber 2. These balls will be visible to the user through the chamber window 3.

The user is also required to select the number of balls that are to be drawn from the chamber 2, and this is done using the draw size selector 6.

Again, a numerical keypad or set of numbered buttons is preferably provided to allow the user to input the draw size.

The user then employs the bet selector 7 to place a bet on an outcome of the drawing of the specified number of balls from the chamber 2. In the present example, the user simply makes a prediction regarding how many red balls and how many black balls will be present in the specified selection from the chamber 2. The order in which the balls is drawn is (for the moment) unimportant. For example, if the user specifies that three balls are to be drawn from the chamber 2, and uses the bet selector 7 to predict that the outcome of the selection from the chamber 2 will be two black balls and one red ball, any of

the following orders of drawn balls will win: black, black, red; black, red, black; or red, black, black Once the user has placed a bet by making a prediction on the outcome of the selection from the chamber 2, the first gaming apparatus 1 provides the user with appropriate odds. In order to do so, the first gaming apparatus 1 must first determine the probability of the user's selection being correct. Preferably, this is carried out using a processor (not shown), operable to receive inputs from the various selectors, to calculate odds (explained in more detail below) and output the odds for presentation to the user via a suitable display.

The computation of selection probabilities is carried out using the probability formula, as given below, for randomly selecting items, where the order of the selection is unimportant (an example of this type of selection is given above), and assuming that the items have equal probability of selection and are not replaced for reselection once selected. The probability of any particular selection may be defined as the ratio of the number of ways that the particular selection may be made to the number of ways that all the selections of the same size as the particular selection may be made.

For the first gaming apparatus 1, the probability of any particular selection (i.e. combination, where the order doesn't matter) is equal to (ucv x pCq)/nCm, where n is the total number of balls in the chamber 2 at the start of a game, m is the total number of balls in the selection, u is the number of black balls in the chamber 2 at the start of a game, v is the number of black balls in the selection, p is the number of red balls in the chamber 2 at the start of a game,

q is the number of red balls in the selection, and aCb equals a!/[(a-b)! x b!], which is the standard combinations formula giving the number of ways of selecting b items from a total of a items, where the order of selection is unimportant. The above probability formula for the first gaming apparatus 1 is the two-colour version of the general probability formula that is applicable to the random selection of any number of different colours. This general probability formula is the product of the combinations formula eCf for each colour in the selection, where e is the number of balls of any particular colour in the chamber 2 for selection at the start of the game and f is the number of balls of that particular colour in the selection, with the total divided by nCm.

For a selection of one ball only from the chamber 2, the probability formula reduces to u/n for one black ball, or pin for one red ball.

When two or more different combinations are combined in a single bet on the outcome of the selection from the chamber 2, then the overall probability is just the sum of the probabilities of each of the individual combinations selected, since these combinations are independent of each other.

The probability figure and any desired house edge are then used to set the odds for any particular combination or combined selection that the user predicts (discussed below). If the probability figure is equal to P. and if traditional odds' or 'fractional odds' are used and have the value "X to Y" [normally written as X-Y, which means that for a stake of Y. a winning bet gives winnings of X and the stake is returned, resulting in a total return of X+Y, and so the ratio of the total return to the stake is (X+Y)/Y, i.e. (X/Y +1)], then for any bet the value of the house edge (in percentage terms, as it is always

referred to herein) is related to these other variables by the following standard formula: house edge (%) = 100 x { 1 - [P x (X/Y +1)]} The corresponding formula for 'decimal odds' and the definition of this type of odds are given below.

The probability figure P could theoretically take on any value if the above ball quantities could take on any values. When realistic restrictions are placed on the ball quantities for practical considerations, as discussed below, then the value of P for the first gaming apparatus 1 may still be set close to any value over a very broad range. For this reason, the first gaming apparatus 1 enables all standard odds values (and other odds values if permitted by a game operator) to be obtained, and for any obtainable odds value the actual house edge may be set close (typically to well within 0.5% in absolute terms) to any value within the full range of zero to 100%.

Examples of some possible games that may be played on the first gaming apparatus 1 are given in tables 1 to 12 appended hereto. The number of black balls and red balls in the first two columns are the numbers of red and black balls in the chamber 2 at the start of the game before any balls have been selected. Table 2 shows examples of different house edge values, ranging from 0.5% to 70%, for the same odds.

The house edge, in percentage terms, is defined as the percentage of the total amount staked by the players that in the long-term goes to the game operators, and this assumes that all of the bets made have a probability figure of P and have the same stake, and that the ratio of the number of winning bets to

the total number of bets placed is equal to P in the long-term. Although this ratio may in actuality differ from P after any specific number of bets for statistical reasons, the ratio ultimately becomes very close to or possibly equal to P as the number of bets increases (and this fact may be used as a definition of the probability figure). In practice, for players placing a variety of types of bets and stakes, then the percentage of the total amount staked that in the long-term goes to the game operators would typically be close to the average house edge per bet, whatever the values of the probability figure and stake for each bet, due to the effects of averaging.

House edge levels are chosen by the game operators by setting target and minimum house edge figures or a combined target/minimum figure, as explained below, and these house edge figures may be set to any values.

As mentioned above, for any obtainable odds value the actual house edge may be set close to any value within the full range of zero to 100%. The actual house edge values for the game examples given herein have been set to lie mostly within the range of 0.9% to 2.9%. However, house edge values in all the given examples cover the range 0.002% to 70%.

Preferably, once the user has selected the number of red and black balls to be introduced into the chamber 2, and the number of balls that are to be drawn from the chamber 2, the first gaming apparatus 1 displays all of the possible outcomes of combinations (or 'lines') that may arise from the selection from the chamber 2. The user selects any number of the possible combinations and the first gaming apparatus 1 will also calculate appropriate odds for the selected outcomes, and these odds are displayed. The user places a stake, and the specified number of balls are then drawn from the chamber 2.

The total number of available combinations or lines is equal to m+l, where m is the selected number of balls to be drawn from the chamber 2 or the length' of the selection line, and so any number up to m of these lines may be selected (this assumes that the number of black balls and the number of red balls in the chamber 2 at the start of a game are each greater than or equal to m, as otherwise the total number of available lines would be reduced). For example, Figure 1 shows a configuration of the first gaming apparatus 1 in which there are 3 balls in the selection (i.e. m = 3) and the 4 (i.e. m+l) available lines, and these available lines are: 3 black balls 3 red balls 2 black balls and 1 red ball 1 black ball and 2 red balls.

For the configuration of the first gaming apparatus 1 shown in Figure 1, the first two of the above available lines have been selected by the player, i.e. the bet is on 3 black balls or 3 red balls coming up.

The drawing of the balls from the chamber 2 may start automatically when all of the required steps have been taken, or may be positively initiated by the user, for instance by the pulling of a handle 8 (in the style of a "one arm bandit") or activation of an alternative starting means.

As discussed above, the balls drawn from the chamber 2 are visible to the user through the viewing window 4, and the user is able to tell whether he or she has won the bet.

A first gaming apparatus l is also provided with an outcome determiner, operable to detect the colour of the balls that have been drawn from the chamber 2, and establish whether the user has won the bet. If the user has won the bet, the appropriate winnings are calculated and dispensed together with the stake for collection by the user, and a skilled person will appreciate how this may be achieved. If the user has failed to win the bet, neither any winnings nor the stake are returned.

As discussed below, maximum and minimum limits may apply for stakes, winnings, and also odds values, and there are various selections for which bets are not available.

In use of the first gaming apparatus 1 described above, only one bet per game is allowed, and each bet is upon all the balls in the selection, and all possible lines are presented from which any number of lines may be selected by the user for the bet, and the order in which the balls are drawn from the chamber 2 is unimportant. This simplicity makes the first gaming apparatus I extremely easy to use.

Turning to Figure 2, a second gaming apparatus 9 is shown. The second gaming apparatus 9 has several features in common with the first gaming apparatus 1, and these features are indicated with the same reference numerals as used in relation to Figure 1.

The main difference between the second gaming apparatus 9 and the first gaming apparatus I is that the second gaming apparatus 9 offers many more betting possibilities. When using the second gaming apparatus 9 it is also possible to bet, for example, that the outcome of a selection of four balls from the chamber 2 will be, say, red, black, red, black in that order. Since the order

matters, this type of bet is called an "exact position bet". When using the second gaming apparatus 9 it is also possible to place a separate bet (called a "side-bet") on, for example, the third and fourth balls being, say, a red and a black, either as an "order doesn't matter" bet or as an exact position bet.

Generally, a side-bet placed on the second gaming apparatus 9 allows the possibility of betting on the outcome of any number (from 1 to m, where m is the number of balls drawn from the chamber 2) of the positions (and any of the positions) within the selection line, as opposed to a "main bet" which is a bet on the outcome of the complete selection from the chamber 2. A side-bet may be either an "order doesn't matter" bet or an exact position bet. The second gaming apparatus 9 also allows several of each of these types of bet to be placed on the outcome of one game.

The procedure for playing a game on the second gaming apparatus 9 is that the player chooses the numbers of black balls and red balls to be placed in the chamber 2, selects the number of balls that are to be drawn therefrom, and then chooses the selections for the main bets and the sidebets. It is possible for the second gaming apparatus 9 to show all possible lines for "order doesn't matter" bets, as for the first gaming apparatus 1, provided that enough room is available on the display. The main bets are all of the same line length, equal to the draw size specified by the user, and any of these main bets may be combined into a single bet (the probability of the combined selection is just the sum of the probabilities of each of the individual lines selected if these predictions are independent of each other. However, if any of the predictions are not independent, but are already included in the other lines, e.g. when an exact position bet is already included in an "order doesn't matter" bet, then the probabilities of the already included predictions are not added on). After each bet is selected, the odds for that bet automatically appear, and the stake is then

chosen and placed. When all the bets have been chosen, the game is initiated as discussed above. As discussed below, maximum and minimum limits may apply for stakes, winnings, and also odds values, and there are various selections for which bets are not available.

The probability calculation relating to the exact position main bets and exact position side-bets is just the product of the probabilities of each ball in the exact position selection being selected in turn, taking into account the fact that there is one fewer ball of a particular colour in the chamber 2 for the selection of each subsequent ball. For example, referring to the probability formulae for one ball selection given above (i.e. P = u/n for one black ball, or P = pin for one red ball), the probability of choosing black, black, red (in that order) is equal to u/n x (u-l)/(n-l) x p/(n-2). The format of the probability formula is in fact the same for the exact position side-bets and for the exact position main bets, and is completely independent of the position of the balls in the side-bets. The reason for this is that the selection for any particular position is as random as that for any other position. So, the probability calculation for any exact position side-

bet is the same as the calculation that would apply if the balls in that exact position side-bet were the first balls to be selected by the machine.

Similarly, the probability formula is the same for the "order doesn't matter" side-bets and for the "order doesn't matter" main bets and is the one used for the "order doesn't matter" bets as discussed above. Again, the probability calculation is completely independent of the position of the balls in these side-bets for the same reason as given above, i.e. the selection for any particular position is as random as that for any other position. So, the probability calculation for any "order doesn't matter" side-bet is the same as the calculation that would apply if the balls in that "order doesn't matter" side-bet were the first balls to be selected by the machine.

Examples of exact position bets are given in Table 13. The examples in this table only apply to the second gaming apparatus 9 and not to the first gaming apparatus 1, whereas all the other examples given herein apply to both types of apparatuses 1,9.

In the gaming apparatuses 1,9 (or machines 1,9), the odds figures determined by the machines would normally be taken from a list of standard betting odds that are in general use, and such a list, or any other odds list or odds format chosen by the game operators, would be incorporated into the machines. There are different lists of standard betting odds available, and, as examples, lists of traditional or fractional odds (this type of odds is used in the U.K. and some other countries) appear in Lists 1 and 2. (In Lists 1 and 2 and elsewhere herein, traditional or fractional odds of X-Y with X larger than Y are referred to as odds-against, and when Y is larger than X the odds are referred to as odds-on (and evens are 1-1). For odds-against figures, as the ratio X/Y becomes larger then the odds-against figures are referred to herein as becoming higher', and for odds-on figures as the ratio Y/X becomes larger then the odds-

on figures are referred to as becoming 'higher'.) Any odds formats, such as decimal odds, may be used if desired by the game operators, and a number of these odds formats are described later.

As well as players being able to set their bets by selecting the various game parameters (i.e. the numbers of black balls and red balls in the chamber 2, the number of balls in the selection, and also the selection or selections that they would like to bet on) as stated above, the first and second gaming apparatuses 1,9 could in addition allow players to choose sets of game parameters from pre-selected lists and lists saved by the players. There could also be an option of a "lucky dip" to randomly select sets of game parameters

from these pre-selected and saved lists. As discussed below, when a bet is not possible for a set of game parameters or for a combination of game parameters and stake, then the machine 1,9 could advise the players how to proceed to a viable bet, or suggest a similar but viable bet, which may be taken from pre-

selected lists. Another possible game option would be for players to choose the odds directly, and the machine 1,9 would then automatically compute an appropriate set of game parameters (there would be several alternative sets for any odds value), or use a pre-selected list to produce these odds. All the pre-

selected lists could be produced automatically as part of a setting up process just after the game operators enter the target and minimum house edge figures or combined target/minimum figure, as discussed below. Alternatively, pre-

selected lists taken from tables similar to the ones that are appended hereto may be used, provided that the range of house edge figures is acceptable to the game operators. As mentioned above, for any obtainable odds value the actual house edge in a game may be set close to any value within the full range of zero to 100%, and so games with relatively low house edge values may be produced across the whole range of odds. Table 10 shows examples of games with a house edge value between 0.002% and 0. 5%. For the selections producing odds of 1-1 and 3-1 given in Table 10, the house edge value decreases quickly as the number of balls increases, with equal numbers of black balls and red balls in the chamber 2 of the apparatus 1,9 at the start, and this allows for easy selection of the house edge within a very broad range for these odds values. In any list of pre-selected game parameters, some games with relatively low house edge values could be included and highlighted in order to attract more players to all of the games.

It is possible to implement cut-down or reduced versions of the games based on options outlined above, especially those options involving pre-

selected lists. For example, one cut-down version involves the players not being able to set the various game parameters directly but being able to choose the odds directly, with a limited number of odds values available on a pre-

selected list and with each odds value corresponding to a pre-selected game (or a choice of pre-selected games if more than one game is made available on the list for any particular odds value). For this type of cut-down version, an example of a list of available odds values for traditional or fractional odds is 1-

1, 2-1, 5-1, 10-1, 25-1, 50-1, 100-1, 500-1, 1,000-1, 10,000-1, 100,000-1, and 1,000,000-1. An example of a fuller list of available odds values for this type of cut-down version is 1-2, 1-1, 6-4, then X-1 with X taking on any whole number from 2 to 100, and additionally 250-1, 500-1, 1,000-1, 2,500-1, 5,000- 1, 10,000-1, 25,000-1, 50,000-1, 100,000-1, 250,000-1, 500,000-1 and 1,

000,000-1. This and other cut-down versions of the games could be played using a number of types of machines or operational formats, such as: 1. reduced and specialised versions of any of the apparatuses 1,9 that provide full versions of the games, 2. telephone betting, where the calls could be received by people or by an automated system, with either method using reduced and specialised versions of any of the apparatuses that provide full versions of the games, or 3. scratch-cards and other types of batch implementations.

A scratch-card implementation of the cut-down version of the games described herein could be realised. Each scratch-card would contain one or more games, with the total stake, i.e. the price of the scratch-card, shared between the games. There could be different sets of scratch-cards containing

the same games as each other but with different stakes. The presentation on the scratch-cards could consist of a bold statement of the odds and the stake

together with a basic schematic drawing for each game. The outcome of any game on the scratch-cards could be pre-determined by the same random ball selection process that is described herein for the full versions of the games, and each card would display (once the protective covering has been scratched off) the result of the game, as well as whether the card is a winning card or not.

However, for any batch of scratch-cards it would be possible to check the house profitability of the batch and to make any adjustments to the results before issuing that batch or to decide not to issue that batch.

In a preferred embodiment, the games offered on each batch of scratch-

cards comprise one odds value and one stake value. In this way, a batch of cards comprises games and their outcomes from a predetermined list.

The batch principle as described above may also be applied to computer-

based networked implementations of the games, as mentioned later, or with other games, with each game and its result in a batch taken from a predetermined list. As described above, the outcome of any game in a batch may be predetermined by the same random ball selection process that is described herein for the full versions of the games, and for any batch it would be possible to check the house profitability of the batch and to make any adjustments to the results before issuing that batch or deciding not to issue that batch. This ensures that, once all of the scratch-cards, computer-based games, or other games in a batch have been used, an overall return from the batch would be realised, and this would allow the game operators a degree of control over the financial exposure resulting from the issue of the batch, as well as guaranteeing a certain return to the public.

The house edge values for scratch-cards are usually relatively high and would typically be higher than those for other versions of the games. As previously referred to, for any obtainable odds value the actual house edge in the games may be set close to any value within the full range of zero to 100%, and so games with a house edge value close to any relatively high house edge value may be produced across the whole range of odds. Tables 1 la - c show games with a house edge of 50% and include all the odds values in the first odds list example given in this section and contain games that would be suitable for scratch-cards.

In many scratch-card implementations it would be preferable to have available relatively few odds values over a broad range, such as the eight values 2-l, lO-1, 50-1, 100-1, 1,000-1, 10,000-1, 100,000-1 and 1, 000,000-1, or the ten values 1-1, 5-1, 10-1, 50-1, 100-1, 500-1, 1,000-1, 10,000-1, 100,000-1 and 1,000,000-1. Fewer available odds values would lead to a higher chance that any particular value would win in any community, and the number of wins at any particular odds value is important for causing word of mouth recommendations to play the scratchcard games. For this reason, lower odds values of up to around 1,000-1 are important, but is also important to include higher odds values up to and including, say, 1,000,000-1 in order to provide the attraction of the chance of a large win, and to include a broad range of odds from small to high values so that players have a reasonable choice over the range. It is possible to implement pre-selected games that provide lower levels of winnings for partial success, and three examples of such games are as follows:

1. Top prize $1,000,000. In this game, any of 3, 4, 5, or 6 black balls are required for a win, the house edge is 50%, and the probability of one or more of the balls selected being black (i.e. a winning ball) is greater than 50% (actually 56%). See Table 14 for the prizes that may be won for various selections, based on an initial stake of $1.

2. Top prize $100,000. In this game, any of 3, 4, or 5 black balls are required for a win, the house edge is 50%, and the probability of one or more of the balls selected being black (i.e. a winning ball) is greater than 50% (actually 53%). See Table 15 for the prizes that may be won for various selections, based on an initial stake of $1.

3. Top prize $10,000. In this game, any of 3, 4, or 5 black balls are required for a win, the house edge is 50%, and the probability of one or more of the balls selected being black (i.e. a winning ball) is greater than 50% (actually 55%). See Table 16 for the prizes that may be won for various selections, based on an initial stake of $1.

The prize in the above three games is equal to the winnings plus the stake. The product of the probability figures and prizes for these games has been biased towards the smallest and largest prizes. As mentioned above, the house edge for these games has been set at 50% in each case. As also mentioned, in most cases at least one ball in the selection is a winning ball for each of these games, and so this would help to encourage players to have another go if they did not win.

For these types of games, the product of the probability figure and the ratio of the total return to the stake in the house edge formulae given herein is

replaced by the sum of the product of the probability figures and the prizes for a unit stake. In the above three game examples, the 'probability of winning" figure of 1 in Z is equivalent to a probability figure of 1/Z.

The above three game examples may in some situations be preferable to the three odds values of 1,000,000-1, 100,000-1, and 10,000-1 in the eight and ten odds values examples for scratch-card games as given above. The reason for this is that in any community, wins at these three high odds values are likely to arise very rarely, if at all, as opposed to wins at the odds values of 1,000-1 and lower. As previously mentioned, wins in a community are very important for word of mouth recommendations. For the above three games with 'stepped' winnings, all the prizes except for the top one in each game and the $2,000 prize in the first game would typically be won fairly regularly in any community, and so these games may in some situations be more likely to be played on a regular basis.

As indicated above, in implementation of the first or second gaming apparatuses 1,9 there would need to be maximum limits set by the game operators on the number of black balls and the number of red balls in the apparatus 1,9 and also on the number of balls in the selection. Although the limits may be set at various levels, a limit of 200 balls for each of the two colours, together with a limit of 10 balls for the selection would be adequate to ensure the availability of an extensive range of combinations of odds and house edge values as well as a large variety of available games. All the game examples given herein conform to these limits.

Referring to the house edge formulae given herein, the maximum value of the house edge that would be possible for a given value of the probability figure P is given as follows:

maximum house edge (%) = 100 x (1 - P) This means that for a set of game parameters resulting in a value P that leads to a maximum house edge value below the value set for the minimum or combined target/minimum house edge figure, as referred to below, then no odds could be made available for those game parameters (this is relevant for higher values of P). Reference is made to the situations in which the actual house edge value approaches (and in the limit equals) the maximum house edge value, resulting in the odds-on value becoming increasingly sensitive to changes in the values of the actual house edge and the probability figure P. If the player enters game parameters which, for the reasons given above, result in odds not being possible, then the apparatus 1,9 would not allow a bet with those game parameters. The apparatus 1,9 would of course inform the player accordingly and could advise the player on how to proceed to a viable game, or suggest a similar but viable game, which may be taken from pre-

selected lists, as referred to above.

Odds-against values are increased by decreasing the probability figure P. and so high odds-against figures may be obtained by choosing a number of balls in the selection all with the same colour, as shown in Table 12. The highest odds-against figures are obtained by choosing several balls in the selection, all with the same colour, and having the number of balls in the chamber 2 of the apparatus 1,9 of this colour equal to the number in the selection, and having many balls of the other colour in the chamber 2 of the apparatus 1,9.

Even with the practical limits on the numbers of balls, as given above, extremely high odds-against values, as indicated by Table 12, for example tens or hundreds of millions to one, or very much higher if required, may be generated with the apparatus 1,9 with the type of game parameters described above. However, the game payout would need to be limited by the game operators to below a specified value, as referred to below.

The availability of high odds-against figures provides the chance of high payouts and so adds significantly to the attractiveness of the games offered.

Apart from the scratch-card games referred to above, a networked implementation of the games would be most suitable for providing for the availability of high odds-against values and also for providing for the possibility of larger stakes, since connection to a network would allow constant centralised control and monitoring by the game operators.

Odds-on values are increased by increasing the probability figure P. and this is achieved by selecting more than one of the possible combinations or outcomes, with the odds-on figure increasing as more combinations are selected. High odds-on figures may theoretically be obtained by choosing several balls in the selection and having many balls of both colours in the chamber 2 of the apparatus 1,9 and selecting all of the lines except for one same colour line. To produce the highest odds-on figures, then the number of balls in the chamber 2 of the apparatus 1,9 of the colour of the one unselected same colour line would be set to equal the number of balls in the selection. For higher values of the probability figure P. though, odds may not be available because of the set value of the minimum house edge figure, as detailed above.

Nevertheless, any high odds-on value up to a very high figure may theoretically be obtained using the formulae for odds-on values, irrespective of the set house edge figures, due to the nature of these formulae, as referred to below.

However, although high odds-on values could be used in apparatuses 1,9 with the type of game parameters described above, there are a number of reasons why the size of odds-on values would be restricted: 1) High oddson values are not typically used in betting and, as indicated in Lists 1 and 2, odds-on values do not normally exceed 1-10.

2) A win may be too small to register with high odds-on values, as mentioned below.

3) If the actual house edge figure were not small enough, then a high odds-on value would result in the winnings to stake ratio being relatively small compared to the ratio that would apply for a zero house edge (or, expressing it another way, there would be a relatively small probability of the player showing an overall positive total return after many games).

This is a universal property of odds-on values, irrespective of which betting scheme is used. It is due to the fact that the odds-on value, as given in the house edge formulae contained herein, becomes highly sensitive to changes in the values of the house edge and the probability figure P as the actual house edge value approaches the maximum house edge value, as mentioned above. This effect increases as the actual house edge and the maximum house edge values converge, and in the limit of the two values being equal, then the odds-on value would be infinitely high and so the winnings would be zero.

Due to the last reason given above, an odds-on limiting scheme providing a maximum available odds-on figure would need to be implemented in the offered games. One such scheme would be simply to choose a maximum oddson figure, such as 1-10, which is applicable to all bets, and this figure is then incorporated into the apparatuses 1,9. This scheme would be

straightforward to implement and would ensure that all standard odds-on values up to the chosen maximum were available, whatever the house edge values chosen by the game operators. If the desired house edge values were small, then this maximum odds-on figure could be increased accordingly.

In order to ensure that no win is too small to register, the product of the odds-on winnings to stake ratio and the stake for any bet would need to be limited by the game operators to above a specified value, as referred to below, and this would need to be considered independently of the odds-on limiting scheme. If the maximum odds-on figure were set to 110, as suggested above, and if the ratio of the minimum stake to the smallest possible winnings were 10 or more (e.g. a minimum stake of lOc and smallest possible winnings of to or less), then no win could be too small to register, and all three of the above listed reasons for limiting the size of the odds-on values would have been catered for.

A maximum limit on the total of the stakes for each game, irrespective of the odds, would need to be set by the game operators in order to limit the amount that may be lost by the player in a game. An individual maximum total stake limit may also need to be set for a specific player for a particular game for reasons such as available credit. For practical purposes, a minimum stake level (apart from a zero stake to allow games to be played without betting) would need to be set by the game operators for each bet.

As referred to above, a maximum limit would need to be set by the game operators on the amount that may be won in a game, irrespective of the number of bets in the game, in order to protect the interests of the game operators, and the player would be informed of this winnings limit. So, for a game involving

one bet and for traditional or fractional odds of X-Y, this maximum limit would apply to the product of X/Y and the stake for that bet. If this limit were exceeded for the desired stake, then the player would be advised of this and the value of the stake could be reduced accordingly. If the game winnings limit would still be exceeded with the stake at the minimum level, then the player would be informed of this before playing, allowing the player the opportunity of deciding whether or not to proceed with that particular bet. If the player decided not to continue with the bet, the apparatus 1, 9 could advise ways of proceeding to an acceptable one. For games with more than one bet, then, for ease of implementation, the same stake reduction procedure as above could be used for each bet separately when the maximum winnings limit is exceeded for any bet, although each bet within a game may not be independent. An alternative procedure for games with more than one bet is for the machine to determine which bets are not independent and then adjust the stakes accordingly to the most appropriate levels if the maximum winnings limit is exceeded, and then proceed as above. Whatever procedure is used when the maximum winnings limit is exceeded, it would be made clear to the player that the total winnings on the game, irrespective of the number of bets in the game, were limited to a specific amount.

As explained above, an odds-on limiting scheme is necessary. As also referred to in that section, to ensure that no win is too small to register, a minimum limit would need to be set by the game operators on the amount that may be won in a bet, and so, for traditional or fractional odds of X-Y, this minimum limit would apply to the product of X/Y and the stake for any bet. In the case of a proposed bet where the house edge corresponding to the maximum odds-on figure would be lower than the minimum or combined target/minimum house edge figure, as referred to below, or in the case of the limit on the minimum amount that may be won not being reached, then the proposed bet

would not be permitted, and the machine could advise ways of proceeding to a viable bet.

Various methods for setting a house edge figure for each bet would be possible, but those described here are the most straightforward. The house edge, in the form of a target figure and a minimum figure or, alternatively, a combined target/minimum figure, is entered during the set up procedure by the game operators. The target figure is equal to the desired house edge value, and the minimum figure is equal to the smallest house edge value that would be acceptable to the game operators for any bet. The combined target/minimum figure is equivalent to the separate target and minimum figures being set equal to each other, as referred to below. For traditional or fractional odds, the actual house edge for any particular bet may differ significantly from the target figure or from the combined target/minimum figure, as described below. However, for decimal odds, or for any other odds format that is based on decimal numbers rather than on whole numbers, the actual house edge figure could be very close to the target figure or to the combined target/minimum figure, as also described below. The target and minimum house edge figures or combined target/minimum figure may be set to any values, and so may be as low or as high as desired by the game operators.

A number of mathematical schemes would be possible for determining the value of the traditional or fractional odds together with the actual house edge figure for any bet, but the schemes given below are the most straightforward. The apparatuses 1,9 would determine the odds from the probability and house edge figures, and then display these odds for all bets.

The apparatuses 1,9 enable all standard values of traditional or fractional odds (and other odds values if permitted by the game operators) to be obtained.

As described above, there are some values of the probability figure P for which no odds are available, since the actual house edge value for these values of P can never be equal to or above the minimum or combined target/minimum house edge figure, and the formula given in that section may be used by the apparatuses 1,9 to determine whether odds are available. If they are, then the schemes described below may be used to determine their value.

When separate target and minimum house edge figures are used, one method for the apparatuses 1,9 to compute the traditional or fractional odds for any particular bet would involve using the probability figure for that bet to find the two odds values (from the odds list being used) that result in house edge values just below and just above the target figure (i.e. the largest value below and the smallest value above). Then, the odds with the house edge value closer to the target figure would be chosen. If the lower house edge figure is closer to the target figure but also lower than the minimum figure, then the odds with the higher house edge figure are chosen instead. However, in the event that an odds value is listed which results in a house edge value exactly equal to the target figure, then that odds value would be the one that is used. Extra initial stages may be necessary in some implementations to simplify the process, and these would involve finding the odds value that produces a house edge figure that is just positive (i.e. the smallest positive value), and then finding all the odds from this first value to the value that results in a house edge just above the target figure, and then proceeding as before.

If in the above scheme the determined odds were odds-on and were higher than the maximum limit, as referred to above, then this maximum limit

would be used for the actual odds provided that the actual house edge with these odds were not below the minimum house edge figure. Otherwise, no odds would be made available.

As mentioned above, an alternative set up procedure relating to the house edge would require the game operators to enter just one value, a combined target/minimum house edge figure, rather than the separate target and minimum figures required for the scheme outlined above. The method for the apparatus 1,9 to compute the traditional or fractional odds would be similar to the one described above and would involve finding the odds value (from the odds list being used) that results in a house edge just above, or equal to, this target/minimum figure. This alternative scheme is equivalent to the above scheme with the separate target and minimum house edge figures being set equal to each other. This alternative set up procedure would be slightly more straightforward to carry out, involving entering just one number rather than two. Comparing the results of these two schemes, when assuming that the combined target/minimum figure and the separate minimum figure are equal and the separate target figure is set to a value higher than the other figures, then this alternative procedure would of course result in a lower average actual house edge figure.

If in the above alternative scheme the determined odds were odds-on and were higher than the maximum limit, as referred to above, then no odds would be made available.

Other restrictions on the available odds values, as outlined above, would also need to be taken into account in any odds computation procedure.

For traditional or fractional odds, since these odds can only take on discrete values, the actual house edge figure for any specific bet could be significantly higher than the target figure or the combined target/minimum figure. Ensuring that as full a list of betting odds as possible is used will minimise the occurrence of these significantly higher actual house edge figures.

List 2 is an example of a more complete list (but containing some less common values) as compared to a typical list of traditional or fractional odds in general use in the UK as given in List 1. Adjacent values of odds X-Y in List 1 may have the ratio of (X/Y +1) differing by up to l 1. 1%, resulting in an actual house edge for some bets of up to this sort of magnitude above the minimum figure (rather than above the target figure, and assuming that the target figure is within a few percent above the minimum figure) or above the combined target/minimum figure. For example, if the minimum figure or the combined target/minimum figure were set to 2. 5% (and the target figure was within a few percent above 2.5%), bets could have an actual house edge figure of up to 13.6%.

Irrespective of which list of traditional or fractional odds is used, however, there are always many possible bets where the actual house edge is close to the target figure or to the minimum figure or to the combined target/minimum figure. Players could if they wish calculate the actual house edge themselves to determine the best bets, i.e. those with the smallest house edge figures, or in some instances they could find the best bets by changing the settings of the various game parameters and viewing how the odds change.

Alternatively, as well as displaying the odds, the apparatuses 1,9 could also display the actual house edge value for any bet, if desired by the game operators.

By using an odds format that is based on decimal numbers rather than on whole numbers, as described below, the actual house edge figure could be very close to the target figure or to the combined target/minimum figure.

As previously mentioned, as well as catering for standard traditional or fractional odds it would be possible with the apparatuses 1,9 to provide for any odds formats, including non-standard odds, if desired by the game operators.

For example, X or Y could theoretically be any number, whole or decimal, for odds of X-Y. So, X or Y could be whole numbers that are not on any standard lists of traditional or fractional odds, or, alternatively, X or Y could be decimal numbers, e.g. odds of 9.5-l, 5.25-1, 1.94-1, 1-2.08, or 1-4.5. This decimal number odds format of X- 1 for odds-against and 1 - Y for odds-on is in fact used in some countries, and the US 'money line' format (sometimes called American odds') is really the same as this decimal number odds format although it is presented differently: using the above decimal number odds examples, 9.5-1 is +950 in the 'money line' format, 5.25-1 is +525, 1.94-1 is +194, 1-2.08 is -208, and 1-4.5 is -450. Another type of odds, 'decimal odds' (sometimes called 'European odds'), as opposed to the above-mentioned decimal number odds format, is described below.

Decimal odds give the total return as a multiple of the stake, rather than the winnings to stake ratio of the 'X-Y' formats. For example, decimal odds of 6.00 correspond to traditional or fractional odds of 5-1, 2.00 corresponds to 1 - 1, 1.73 corresponds to 8-11, and 1.53 corresponds to 8-15. So, if the decimal odds are equal to D, then the value of D which corresponds to traditional or fractional odds of X-Y is (X+Y)/Y, i. e. D = X/Y +1. This decimal odds format, which is preferred in a number of countries, is generally more straightforward for players to understand, especially for beginners to betting, and particularly for odds-on values. These decimal odds (and also the decimal numbers in the

decimal number odds format referred to above) could take on any value limited only by the number of decimal places allowed and so approach being a continuous variable (as opposed to a discrete variable), and so when this type of odds format is used the actual house edge figures could be set very close to the target figure or to the combined target/minimum figure.

When decimal odds are used, the house edge formula becomes: house edge (%) = 100 x [ 1 - (P x D)] The method for the apparatuses 1,9 to determine decimal odds values is straightforward, and involves calculating D in the above equation using the probability figure P for the chosen bet, and where the house edge is the target figure or the combined target/minimum figure, and then rounding down the resultant value of D to the required number of decimal places.

A minimum house edge figure is still needed for decimal odds for some higher values of P to determine when no odds are available, and also when the odds-on figure corresponding to the target house edge figure is larger than the set maximum, as referred to above. In fact, due to the nature of the number system for decimal odds, this 'maximum' odds-on figure when referring to traditional or fractional odds is actually a minimum decimal odds figure below which no decimal odds values would be permitted, e.g. a maximum odds-on figure in traditional or fractional odds of 1-10 corresponds to a minimum decimal odds figure of 1.10. So, if the determined decimal odds were lower than the minimum figure, then this minimum figure would be used for the actual odds provided that the actual house edge with these odds were not below the minimum house edge figure. Otherwise, no odds would be made available. Other restrictions on the available odds values, as outlined above, would

also need to be taken into account in any odds determination procedure.

It would of course be possible for more than one type of odds format to be displayed at the same time by the apparatuses 1,9, if desired by the game operators. For example, when traditional or fractional odds are used, the apparatuses 1,9 could display the decimal odds as well. When these two types of odds are displayed together, the value of the traditional or fractional odds would need to be calculated first, and then the calculation of the decimal odds would be made using the formula D = X/Y + 1 (and so the actual house edge would be dependent on the traditional or fractional odds). This option of displaying the decimal odds together with the traditional or fractional odds would provide odds figures that are more readily understandable in a number of countries, and could also be a useful feature for beginners to betting because of the simplicity of the decimal odds format.

For a software implementation of the apparatuses 1,9 or for an electronic simulation, there are various methods that may be used to produce the algorithm that determines that the selection of balls is done on a random basis.

The algorithm must of course take into account that once a ball has been selected it is not returned to the chamber 2 for re-selection during that game.

The most straightforward method uses the assumption that the balls are selected in sequence, as opposed to some or all of the balls in the selection line being selected at the same time (although the actual selection method used, as long as it is random, does not affect the probability of any particular outcome). Then, the calculation of the probabilities for the selection of each ball in turn depends on the number of balls in the chamber 2 of the apparatus 1,9 just before a particular ball is selected. Referring to the probability formulae for one ball

selection given above, these formulae are P = u/n for the first ball in the selection being black and P = pin for this first ball being red. Assuming, say, that the apparatus 1,9 chooses red for this first ball, then for the selection of the next ball the probability formula becomes P = u/(n-l) for this next ball being black, and P = (p-l)/(n-1) for it being red, and so on.

One way of setting up the algorithm to ensure that the above probability figures are accurately implemented for the selections of the apparatus 1, 9 would be to designate each ball in the chamber 2 of the apparatus 1,9 with a number, say 1 to n initially, and then 1 to (n-1) after the first ball has been selected, etc. For the balls designated 1 to n initially, the numbers 1 to u could correspond to black balls and so the numbers (u+ 1) to n would correspond to red balls. Then, the program would, using a random-numbers generator or random selector, choose one of these n numbers at random for the selection of the first ball, with each of the n numbers having an equal chance of selection. The same procedure would be followed for the selection of subsequent balls, with the number designations of the balls depending on which ball or balls had been previously selected, e.g. if the first ball chosen were red, then the number designations before the next selection would be 1 to (n-1), with the numbers 1 to u corresponding to black balls and the numbers (u+1) to (n-1) corresponding to red balls, and so on. This method is of course a direct analogy of the random selection of real balls, with each ball being selected in turn.

Although, in the above embodiments, two sets of distinguishable objects are used, the invention is not limited to this, and any higher number of sets may be used. For instance, the first or second gaming apparatus 1,9 could operate with red, black, blue and yellow balls, and a skilled person will appreciate how the apparatuses 1,9 could be adapted for use with higher numbers of sets of

distinguishable objects. Indeed, the probability formula for any number of sets is given above.

As mentioned above, the first and second gaming apparatuses 1,9 need not be mechanical in nature, and may alternatively be computer-based. In this embodiment, a monitor or other display takes the place of the chamber 2 and chamber window 3. The user selects a number of black balls and a number of red balls to go into a "pool" in the same way as described above, and the display shows a computer-generated representation of the selected number of black and red balls mixing together, as if in a chamber. The computer-based gaming apparatus then draws a selected number of balls, which number is input by the user, at random from the balls in the pool (algorithms for this are given above), and the display shows a representation of the drawings of balls of appropriate colour from the pool. A separate display may be provided, in place of the viewing window 4, to show a representation of the balls that have been drawn. In this embodiment, the balls are treated as virtual objects, but the gaming apparatus otherwise operates in the same manner as the above-

described mechanical gaming apparatuses 1,9.

The simplicity of the above games, their visual and playing appeal, and also their versatility, particularly in respect of the availability of an extensive range of odds values and games, will make them attractive to beginners to betting as well as to those who are experienced at betting. In addition, the format and appearance of the games are such that players will get an intuitive indication of the odds involved for each game, as well as being shown the actual odds, and this will give the players more confidence in placing bets.

Also, in some of the games, the perceived odds may appear to be advantageous to the player, and so players may be more inclined to place bets.

An important and attractive feature of the games concept is the ability of players to be able to effectively customise or personalise their games by setting the various game parameters (i.e. the numbers of black balls and red balls in the chamber 2 of the apparatus 1,9 and the number of balls in the selection, as well as the selection or selections that they would like to bet on). Also, the possibility of having options for setting up games, such as choosing from a list of games or choosing odds values, is an important property of the games concept. The possibility of cut-down versions of the games, such as not setting the various game parameters directly but choosing the odds directly from a limited number of available odds values each corresponding to one or more pre-

selected games, is a useful feature and would allow games to be played by using reduced and specialised versions of any of the machines 1,9 that provide full versions of the games, including through telephone betting, or by using scratch-cards or other types of batch implementations. These scratch-cards would include a basic schematic drawing for each game, and would, with their clearly stated odds and stake, enable the players to know exactly what the potential winnings were and would also provide the players with an indication of the probability of winning for any garne. Another useful feature is the possibility of pre-selected games that have the option of being implemented in a batch and which provide "stepped" winnings, i.e. lower levels of winnings for partial success. These games with stepped winnings nicely complement the games with one odds value.

In the present specification "comprises" means "includes or consists of"

and "comprising" means "including or consisting of,'.

The features disclosed in the foregoing description, or the following

claims, or the accompanying drawings, expressed in their specific forms or in terms of a means for performing the disclosed function, or a method or process for attaining the disclosed result, as appropriate, may, separately, or in any combination of such features, be utilised for realising the invention in diverse forms thereof.

List 1 Tvoical list of traditional or fractional odds in General use in the UK 1-1, 11-10, 6-5, 5-4, 11-8, 6-4, 13-8, 7-4, 15-8, 2-1, 9-4, 5-2, 11-4, 3-1, 100-30,

7-2, 4-1, 9-2, 5-1, 11-2, 6-1, 13-2, 7-1, 15-2, 8-1, 9-1, 10-1, 11-1, 12

1, for each subsequent odds value it is most straightforward to use X-1 with X increasing by 1 each time, although only some of these higher value odds-against figures are commonly used.

Odds-on figures are the reverse of the odds-against figures in the above list, although not all of the following odds-on figures are commonly used: 10-11, 5-6, 4-5, 8-11, 4-6, 8-13, 4-7, 8-15, 1-2, 4-9, 2-5, 4-11, 1- 3, 30-100, 2-7,

1-4, 2-9, 1-5, 2-11, 1-6, 2-13, 1-7, 2-15, 1-8, 1-9, 1-10 typical odds-on figures do not exceed 1-10.

List 2 A more complete list of traditional or fractional odds There are several lists of possible traditional or fractional odds that are more complete but contain some less common values as compared to List 1, and an example of one such list is as follows: 1-1, 11-10, 6-5, 5-4, 13-10, 11-8, 7-5, 6-4, 8-5, 13-8, 17-10, 7-4, 9-5, 15-8, 19-

10, 2-1, 21-10, 11-5, 9-4, 23-10, 12-5, 5-2, 13-5, 11-4, 14-5, 3-1, 16-5, 10-3, 7-

2, 4-1, 9-2, 5-1, 11-2, 6-1, 13-2, 7-1, 15-2, 8-1, 17-2, 9-1, 10-1, 11-1, 12-1,

etc. Again, the odds-on figures are the reverse of the odds-against figures in the above list up to, generally, 1-10, although only some of these odds-on figures are commonly used.

Table 1 Examples of names No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 50 50 2 balls selected - both same colour wins (i.e. l-1 1.0 either 2 black balls or 2 red balls win) 100 100 3 bails selected - all three same colour wins (i.e. 3-1 1.5 either 3 black balls or 3 red balls win) 20 20 4 balls selected - 2 black balls and 2 red balls 6-4 1.2 win (order doesn't matter) 5 5 4 balls selected - 2 black balls and 2 red balls 1-1 4.8 win (order doesn't matter) 5 5 6 balls selected - 3 black balls and 3 red balls 1-1 4.8 win (order doesn't matter) 49 100 1ballselected - blackwins 2-1 1.3 33 101 1 ball selected - black wins 3-1 1.5 25 102 1 ball selected - black wins 4-1 1.6 20 102 1 ball selected black wins 5-1 1.6 16 97 lballselected - blackwins 6-1 0. 9 14 99 1 ball selected - black wins 7- 0.9 12 97 1 ball selected - black wins 8-1 0.9 11 100 1 ball selected - black wins 9 1 0.9 10 101 1 ball selected - black wins 10-1 0.9 100 1 ball selected - black wins 1 1 1 0.9 97 1 ball selected - black wins 12-1 1.0 1 0 1 1 ball selected - black wins 20-1 0.9 101 1 ball selected - black wins 25-1 1.0 100 1 ball selected - black wins 33-1 1.0

l O 1 1 ball selected - black wins 50-1 1.0 101 1 ball selected - black wins 100-} 1.0 100 52 1 ball selected - black wins I-2 1.3 99 26 1 ball selected - black wins 1 4 1.0 lOO 11 1ballselected - blackwins 1 10 09

Table 2 Examples of names with different house edge values for the same odds and 1 ball selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 18 181 1 ball selected - black wins 10- 1 0.5 91 1 ball selected - black wins 10- 1 1.0 92 1 ball selected - black wins 10-1 2.0 12 127 lballselected - blackwins 10-1 5.0 56 1 ball selected - black wins 10-1 10 64 1 ball selected - black wins 10- 1 20 21 l ball selected black wins 10-1 50 178 1 ball selected - black wins 10- 1 70 99 100 1 ball selected - black wins 1 - 1 0.5 49 50 lballselected- blackwins 1-1 1.0 49 51 1 ball selected - black wins 1- 1 2.0 19 21 1 ball selected - black wins 1-1 5.0 50 61 1 ball selected - black wins 1-1 10 2 lballselectedblackwins 1-1 20 1 ball selected - black wins 1-1 50 10 57 1 ball selected - black wins 1 - 1 70

Table 3 Examples of names with a house edge of 10% and 1 ball selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 1 ball selected - black wins 1-10 10 2 1 ball selected - black wins 1-2 10 9 11 1 ball selected - black wins 1-1 10 1 ball selected - black wins 2-1 10 17 1 ball selected - black wins 5-1 10 45 1 ball selected - black wins 10- 1 10 67 1 ball selected - black wins 20-1 10 111 1 ball selected - black wins 5 0- 1 1 0 1 1 1 1 ball selected - black wins 100-1 10

Table 4 Examples of games with 2 or more balls selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 50 50 2 balls selected - 2 black balls win 3-1 1.0 50 50 2 balls selected - 2 red balls win 3-1 1.0 100 100 3 balls selected - 3 black balls win 7-1 1.5 47 50 3 balls selected - 3 black balls win 8 1 1.0 44 50 3 balls selected - 3 black balls win 9 1 1.2 44 53 3 balls selected - 3 black balls win 10-1 1. 2 10 10 4 balls selected - 3 black balls and 1 red 1-1 0.9 ball OR 1 black ball and 3 red balls win (order doesn't matter) 10 10 4 balls selected - 3 black balls and 1 red 3-1 0.9 ball win (order doesn't matter) 49 40 4 balls selected - 1 black ball and 3 red 4-1 0.9 balls win (order doesn't matter) 20 20 5 balls selected - 3 black balls and 2 red 2-1 1.2 balls win (order doesn't matter) 30 30 6 balls selected - 3 black balls and 3 red 2-1 1.2 balls win (order doesn't matter) 100 100 7 balls selected - 5 black balls and 2 red 5-1 2.1 balls win (order doesn't matter) 50 46 7 balls selected - 2 black balls and 5 red 6-1 1.4 balls win (order doesn't matter) 34 34 7 balls selected - 6 black balls and 1 red 20-1 0.9 ball win (order doesn't matter) 60 60 8 balls selected - 4 black balls and 4 red 5-2 1.0

balls win (order doesn't matter) = 15 15 balls selected - 6 black balls and 2 red 10-1 1.2 balls win (order doesn't matter)

Table 5 Examples of odds-on names with 2 or more balls selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 20 20 9 balls selected - 5 black balls and 4 red 4-5 1.1 balls OR 4 black balls and 5 red balls win (order doesn't matter) 20 20 7 balls selected - 4 black balls and 3 red 4-6 1.2 balls OR 3 black balls and 4 red balls win (order doesn't matter) 12 12 7 balls selected - 4 black balls and 3 red 4-7 1.1 balls OR 3 black balls and 4 red balls win (order doesn't matter) 20 20 5 balls selected - 3 black balls and 2 red 1-2 1.2 balls OR 2 black balls and 3 red balls win (order doesn't matter) 9 9 5 balls selected - 3 black balls and 2 red 2-5 1.2 balls OR 2 black balls and 3 red balls win (order doesn't matter) 7 7 5 balls selected - 3 black balls and 2 red 1-3 2.1 balls OR 2 black balls and 3 red balls win (order doesn't matter) 10 10 3 balls selected - 2 black balls and 1 red ball 1-4 1.3 OR 1 black ball and 2 red balls win (order doesn't matter) 6 6 3 balls selected - 2 black balls and 1 red ball 1-5 1.8 OR 1 black ball and 2 red balls win (order

doesn't matter) = 5 5 3 balls selected - 2 black balls and I red ball 1 6 2.8 OR 1 black ball and 2 red balls win (order doesn't matter) 4 4 3 balls selected - 2 black balls and 1 red ball 1-7 2.0 OR 1 black ball and 2 red balls win (order doesn't matter) 100 100 3 balls selected - 2 black balls and 1 red ball 1-8 1.4 OR 1 black ball and 2 red balls (order doesn't matter) OR 3 black balls win 20 20 3 balls selected - 2 black balls and 1 red ball 1-9 1.7 OR 1 black ball and 2 red balls (order doesn't matter) OR 3 black balls win 3 3 3 balls selected - 2 black balls and 1 red ball 1-10 1.0 OR 1 black ball and 2 red balls win (order doesn't matter)

Table 6 Examples of higher value odds-aaainst games with 2 black balls selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 20 2 balls selected - 2 black balls win i0-1 2.5 14 2 balls selected - 2 black balls win 20-1 2.2 4 16 2 balls selected - 2 black balls win 30-1 2. 1 17 2 balls selected - 2 black balls win 33-1 2.9 4 19 2 balls selected 2 black balls win 40-1 2.8 10 2 balls selected - 2 black balls win 50- 1 2.2 10 65 2 balls selected - 2 black balls win 60-1 1.1 25 2 balls selected - 2 black balls win 66-1 1.0 26 2 balls selected - 2 black balls win 70-1 2.1 4 28 2 balls selected - 2 black balls win 80-1 2.0 30 2 balls selected - 2 black balls win 90-1 2.7 2 balls selected - 2 black balls win 100-1 1.1

Table 7 Examples of higher value odds-aeainst names with 3 black balls selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 14 16 3 balls selected - 3 black balls win 10-1 1.4 14 23 3 balls selected - 3 black balls win 20-1 1.6 15 3 balls selected - 3 black balls win 30-1 2.0 3 3 balls selected - 3 black balls win 33-1 2.9 12 27 3 balls selected - 3 black balls win 40-l 1.3 12 30 3 balls selected - 3 black balls win 50-l 2.3 20 56 3 balls selected - 3 black balls win 60- 1 1.1 12 3 balls selected - 3 black balls win 66-1 1.5 22 3 balls selected - 3 black balls win 70-1 2.1 10 30 3 balls selected - 3 black balls win 80-l 1.6 28 3 balls selected - 3 black balls win 90-1 1.6 10 33 3 balls selected - 3 black balls win 100-1 1.8

Table 8 Examules of higher value odds-aasinst games with 4 black balls selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 10 4 balls selected - 4 black balls win 10 1 2 9 12 12 4 balls selected 4 black balls win 20-1 2.2 19 24 4 balls selected - 4 black balls win 301 2.6 10 12 4 balls selected - 4 black balls win 33-1 2.4 4 balls selected - 4 black balls win 40-1 2.4 4 bans selected - 4 black balls win 50-1 1.9 11 17 4 balls selected - 4 black balls win 60-1 1.7 10 4 balls selected 4 black balls win 66-1 1.5 11 18 4 balls selected - 4 black balls win 70-1 1.4 13 23 4 balls selected - 4 black balls win 80- 1 1.7 13 24 4 balls selected - 4 black balls win 90-1 1.5 13 25 4 balls selected - 4 black balls win 100-1 2.2

Table 9 Examples of higher value odds-aaainst games with lust black balls selected, and with more black balls than red balls in the chamber 2 of an apparatus 19 at the start No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 15 5 balls selected - 5 black balls win 10-1 1 8 14 10 5 balls selected - 5 black balls win 20-1 1.1 13 6 balls selected - 6 black balls win 30-1 2.0 9 7 5 balls selected - 5 black balls win 33-1 1. 9 5 balls selected - 5 black balls win 40-1 2.4 23 19 6 balls selected - 6 black balls win 50-1 1.9 40 37 6 balls selected - 6 black balls win 60- 1 1.2 30 28 6 balls selected - 6 black balls win 66-1 1.7 18 16 6 balls selected - 6 black balls win 70-1 2.0 21 20 6 balls selected - 6 black balls win 80-1 2.2 16 15 6 balls selected - 6 black balls win 90-1 1.0 13 12 6 balls selected - 6 black balls win 100-1 2.1

Table 10 Examples of names with a house edge value between 0.002% and 0. 5% No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 199 20 1 ball selected - black wins 1-10 0.05 13 13 4 balls selected - 3 black balls and 1 1-1 0.5 red ball OR 1 black ball and 3 red balls win (order doesn't matter) 28 28 4 balls selected - 3 black balls and I 1-1 0. 1 red ball OR 1 black ball and 3 red balls win (order doesn't matter) 87 87 4 balls selected - 3 black balls and 1 1-1 0.01 red ball OR 1 black ball and 3 red balls win (order doesn't matter) 195 195 4 balls selected - 3 black balls and 1 1-1 0.002 red ball OR 1 black ball and 3 red balls win (order doesn't matter) 195 195 4 balls selected - 3 black balls and 1 3-1 0.002 red ball win (order doesn't matter) 32 73 2 balls selected - 2 black balls win 10-1 0.07 14 3 balls selected - 3 black balls win 20- 1 0.4 17 21 4 balls selected - 4 black balls win 30-1 0.05 10 86 2 balls selected - 2 black balls win 100- 1 0.3 20 171 3 balls selected - 3 black balls win 1,000- 1 0.2 63 3 balls selected - 3 black balls win 5,000-1 0. 2 65 3 balls selected - 3 black balls win 50,000-1 0.2 122 4 balls selected - 4 black balls win 10,000,000- 1 0.1

Tables 11a-c Examples of names with a house edge value of 50% (these types of names would be suitable for scratch-cards) No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 150 150 10 balls selected - 5 black balls and 5 1-1 50 red balls win (order doesn't matter) 2 balls selected - 2 black balls win 2-1 50 11 2 balls selected - 2 black balls win 5-1 50 3 9 2 balls selected - 2 black balls win 10-1 50 10 59 2 balls selected - 2 black balls win 25-1 50 59 2 balls selected - 2 black balls win 50-1 50 8 99 2 balls selected - 2 black balls win 100- 1 50 11 3 balls selected - 3 black balls win 500-1 50 57 3 balls selected - 3 black balls win 1,000-1 50 5 102 3 balls selected - 3 black balls win 10,000-1 50 104 3 balls selected - 3 black balls win 100,000-1 50 121 4 balls selected - 4 black balls win 1,000,000-1 50 Table11 b - in these examples of lust black balls selected. the probability of 1 or more of the balls selected being black is Greater than 50% in each case Number of Number Selection Odds House Black Balls Red Balls Edge (%) 100 100 2 balls selected - 2 black balls win 1-1 50 2 balls selected - 2 black balls win 2-1 50 5 11 2 balls selected - 2 black balls win 5-1 50 3 balls selected - 3 black balls win 10-1 50 19 3 balls selected - 3 black balls win 25-1 50 18 3 balls selected - 3 black balls win 50-1 50 14 3 balls selected - 3 black balls win 100-1 50 10 4 balls selected - 4 black balls win 500-1 50

10 48 4 balls selected - 4 black balls win � 1,000-1 � 50 10 5 balls selected - 5 black balls win 10,000-1 50 27 5 balls selected - 5 black balls win 10O,000-1 50 42 6 balls selected - 6 black balls win 1,000,0001 50 Table 11c - in these examples of lust black balls selected. the number of black balls in the machine at the start is Greater than 50 in each case. and the probability of 1 or more of the balls selected being black is Greater than 50% in each case No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 200 200 2 balls selected - 2 black balls win 1-1 50 138 200 2 balls selected - 2 black balls win 2-1 50 156 200 3 balls selected - 3 black balls win 5-1 50 112 200 3 balls selected - 3 black balls win 10-1 50 74 200 3 balls selected - 3 black balls win 25- 1 50 55 199 3 balls selected - 3 black balls win 50-1 50 73 198 4 balls selected - 4 black balls win 100-1 50 69 200 5 balls selected - 5 black balls win 500-1 50 58 200 5 balls selected - 5 black balls win 1,000-1 50 67 199 7 balls selected - 7 black balls win 10,000-1 50 58 196 8 balls selected - 8 black balls win 100,000-1 50 64 194 10 balls selected black balls win 1,000,000-1 50

Table 12 Examples of high value odds-aasinst names No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 4 20 3 balls selected - 3 black balls win 500-1 1.0 26 3 balls selected - 3 black balls win 1,000-1 1.4 80 3 balls selected - 3 black balls win 5,000-1 2 3 81 3 balls selected - 3 black balls win 10,000-1 2.3 5 141 3 balls selected - 3 black balls win 50,000-1 1.6 3 83 3 balls selected - 3 black balls win 100,000-l 2.3 143 3 balls selected - 3 black balls win 500,000- 1 1.6 68 4 balls selected - 4 black balls win 1,000,000-1 2.8 184 4 balls selected - 4 black balls win 10,000,000-1 2.9 5 101 5 balls selected - 5 black balls win 100,000,000-1 1.3 162 5 balls selected - 5 black balls win 1,000,000,000-1 1.9

Table 13 Examples of higher value odds-aesinst shames with exact position bets and 3 balls selected No. of No. of Selection Odds House Black Balls Red Balls Edge (%) 3 5 3 balls selected - 2 black balls and 1 red ball 10-1 1.8 win, and order counts (e.g. specify red, black, black to win) 23 68 3 balls selected - 2 black balls and 1 red ball 20-1 0.9 win, and order counts (e.g. specify red, black, black to win) 6 23 3 balls selected - 2 black balls and 1 red ball 30-1 2.4 win, and order counts (e. g. specify red, black, black to win) 5 20 3 balls selected - 2 black balls and 1 red ball 33-1 1.4 win, and order counts (e.g. specify red, black, black to win) 10 47 3 balls selected - 2 black balls and 1 red ball 40-1 1.2 win, and order counts (e.g. specify red, black, black to win) 5 26 3 balls selected - 2 black balls and 1 red ball 50-l 1.7 win, and order counts (e.g. specify red, black, black to win) 5 29 3 balls selected - 2 black balls and 1 red ball 60-1 1.5 win, and order counts (e. g. specify red, black, black to win) 4 24 3 balls selected - 2 black balls and 1 red ball 66-1 1.8 win, and order counts (e.g. specify red, black,

black to win) 4 25 3 balls selected - 2 black balls and 1 red ball 70-1 2. 8 win, and order counts (e.g. specify red, black, black to win) 4 27 3 balls selected - 2 black balls and 1 red ball 80-1 2.7 win, and order counts (e.g. specify red, black, black to win) 5 37 3 balls selected - 2 black balls and 1 red ball 90-1 2.2 win, and order counts (e.g. specify red, black, black to win) 5 39 3 balls selected - 2 black balls and 1 red ball 100-1 0.9 win, and order counts (e.g. specify red, black, black to win)

Table 14

No. of No. of Selection Prize Probability of Black Balls Red Balls for $1 stake Winning 7 50 6 balls selected - any 3 blacks wins $ 10 1 in 53 - any 4 blacks wins $50 1 in 846 - any 5 blacks wins $2,000 1 in 34,560 - 6 black balls win $1,000,000 1 in 5,184,036 Table 15

No. of No. of Selection Prize Probability of Black Balls Red Balls for $ 1 stake Winning 5 33 5 balls selected - any 3 blacks wins $20 1 in 95 - any 4 blacks wins $275 1 in 3,042 - 5 black balls win $100,000 1 in 501, 942 Table 17

No. of No. of Selection Prize Probability of Black Balls Red Balls for $ 1 stake Winning 10 60 5 balls selected - any 3 blacks wins $12 1 in 57 - any 4 blacks wins $80 1 in 961 - 5 black balls win $10,000 1 in 48,028