CROSS REFERENCE TO RELATED APPLICATIONS

This is a continuation of application Ser. No. 11/299,132 filed on Dec. 12, 2005, abandoned as this one is filed. This application uses the framework of my application Ser. No. 08/691,944, filed on Aug. 5, 1996, now U.S. Pat. No. 5,795,226, granted on Aug. 18, 1998. The inventor's name was misprinted as Chen Yi. A certificate of correction was issued on Nov. 24, 1998.
BACKGROUND OF THE INVENTION

1. Field of Invention

This invention relates to games of chance, more specifically, to methods of playing a betting race game determined by multiple rounds of random numbers.

2. Prior Art

As far as playing surface is concerned every game with a plurality of racers is prior art. As far as betting is concerned, any game of chance, such as keno, lottery, roulette, bingo is prior art. As far as technology is concerned games requiring bet slips and computer data processing such as those at racetrack or shown in patents listed in the Information Disclosure Statement are prior art. But no known race game has allowed bettors to scratch a racer and to assign own preferred finish line for each racer, let alone simply by means of a bet slip. There is no gaming operation generating random numbers to determine racer advancements of various race courses simultaneously.

3. Objects and Advantages

The initial object is to improve the game presented in U.S. Pat. No. 5,795,226. Its nonautomatic version requires a 8′ by 8′ table to set up a race course with betting sections, an exquisite rolling dice box to generate random numbers, and so on, all madetoorder only. Its operation requires several workers. All this means high cost which will result in high house edge. Besides, there is always only one fixed race running for all players to bet on.

This improvement gets rid of any playing table by printing the race course on a bet slip.

This improvement allows a player to start a race anytime with own preferred number of racers and preferred finish lines and to place all kinds of bets just as in U.S. Pat. No. 5,795,226.

This improvement allows a game operator to generate random numbers draw by draw for all racers in any individually set up race course to advance. It also allows a player to determine which upcoming draws of random numbers to be effective on the own setup race so that makeup bets can be placed at leisure between two rounds of advancements or between two races.

Casinos have taken advantage of computer technology to bring out a variety of slots/videogame machines. But basically they are only attractive to really simpleminded people. Sophisticated players never touch them because they are all similar to poker or blackjack while hidden random outcomes may be manipulated despite under government ruling and testing control.

Mutuel handles show that nowadays horse players are fond of exotic bets like pick 3, pick 4. Holding a hanging ticket, better players place makeup bets which can result in sure net win regardless of any following race outcome. But no matter how skillful plus insider information, the high 20% average takeout makes every experienced player ultimately a helpless loser.

At a racetrack the operator doesn't care who win and how much. The situation changes when it is nonparimutuel. Any payoff is operator's loss. Every big hanging bet is a burden which the operator wants to reduce. In order to avoid an eventual huge payoff the operator should allow players to give up a certain percentage of a hanging bet in exchange for credit to place makeup bets that may result in a sure profit of less amount.

Thus, it is time for the gaming industry to provide a low house edge nonparimutuel race game that attracts today's sophisticated players. This invention requires easily madetoorder equipments or existing keno/lottery facilities with minor changes. It's extremely low operation cost allows attractively low house edge.
SUMMARY OF THE INVENTION

The invention provides a race game with a race course as playing surface printed on a bet slip. On a bet slip, besides wagering selections, a player must set up a race by marking one finish line for each of, for example, nine racers. Any racer without marked finish line will be scratched from the race. In order to place makeup bets at leisure between two racer advancements or between two races, a player also selects a draw interval to determine when an upcoming random number draw shall be effective on the own setup race. There are wagering machines connected to a computer to examine bet slips, to print bet tickets, to store and process wagering and all necessary racing data, inclusive payoffs.

The game requires a random number generator such as a set of nine keno bowls—one bowl per racer—, each containing equally many, say, five copies of balls numbered 1 to 6. An operator in charge will use it to produce an ordered set of nine random numbers per draw, to display them on monitors and input into the computer. After a draw, the computer and each player advances racers of the race on bet ticket according to the corresponding random numbers. Any bet ticket holder can let a wagering machine print out the progress of an own setup race. The operator doesn't need to pay attention to the progress of wagering or racing, just produces another draw of random numbers after a regulated period of time.

A multirace bet becomes or remains hanging if it contains a winner in the last race and thus has a chance to be a winner at the end. A hanging bet holder has the option to give up a certain percentage of the original bet in exchange for placing free makeup bets, called credit bets.

Probability formulae as well as how to calculate payoffs and credits will be provided.

The invention also includes an automatic video/computer version of the game.
DRAWINGS

FIG. 1 is a flowchart illustrating the game process.

FIG. 2 is a 1race Win/Place/Show bet slip.

FIG. 2A is a 1race Win/Place/Show bet ticket.

FIG. 3 is a 1race Win/Exacta/Tricta bet slip.

FIG. 3A is a 1race Win/Exacta/Tricta bet ticket.

FIG. 4A is an updated race course after the 1st round of advancements.

FIG. 4B is an updated race course after the 2nd round of advancements.

FIG. 4C show the progress of a race.

FIG. 5 is a multirace Win/Place/Show bet slip.

FIG. 5A is a multirace Win/Place/Show bet ticket.

FIG. 5B is a twice revised multirace Win/Place/Show bet ticket.

FIG. 6 is a multirace Win/Exacta/Tricta bet slip.

FIG. 6A is a multirace Win/Exacta/Tricta bet ticket.

FIG. 6B is a twice revised multirace Win/Exacta/Tricta bet ticket.

FIG. 7 shows a display of random numbers.

FIG. 8 shows an account/activity statement.

FIGS. 9A, 9B, 9C are Win/Place/Show probability and payoff tables.

FIGS. 10A, 10B, 10C are Win/Place/Show probability and payoff tables.

FIGS. 11A, 11C are Tricta probability tables.

FIG. 11B is an Exacta probability table.

FIG. 12A is a line graph to show house edge formula e=2+x/2 for 0<x<=5 and e=4.5+(x−5)/10 for 5<x<=10 where x is the inverse of winning probability.

FIG. 12B is a line graph to show house edge formula e=4+(n+1)*n/2+(n+1)*x/10̂(n+1) for x<10 with n the unique integer satisfying 10̂n<x=<10̂(n+1) where x is the inverse of winning probability.
DESCRIPTION OF THE PREFERRED PLAYING SURFACE AND THE RACE

The invention provides a race course 10 as playing surface printed on a bet slip as shown in FIGS. 2, 3, 5 and 6. The race course contains seventeen numbered strips 11. Strip 0 is where all racers 12 are located to start. Lying in front of every racer is one circle in each strip. All and only those circles lying straight ahead of a racer form the racetrack for that racer to advance. The player marks to select one circle for each racer as finish spot/line. Any racer without a marked circle in front will be scratched from the race.

There are wagering machines connected to a computer to examine bet slips and issue bet ticket as shown in FIGS. 2A, 3A, 5A and 6A. There is a random number generator to produce an ordered set of nine numbers per draw. When a bet ticket is issued, the immediately following draw will be effective on racers of a 1race ticket and on racers of Race 1 in a multirace ticket. The draw number to start the race will be printed on the ticket. Racers advance one after the other, staring from racer 1, as many spots as random number indicating. A random number has no effect if its corresponding racer is scratched. Which one of the following draws will cause the next round of advancement depends on the type of a bet and/or player's preference marked on the bet slip. In the case of multibet, every draw following the first round of advancements causes the next round of advancements until the end of that race. For a 1race bet, the player marks to select a number for draws between two advancements, if, for example, 30 is selected, and if the draw number to start the race is n, then the first advancement takes place at Draw #n, the second advancement takes place at Draw #(n+30), and the third advancement takes place at Draw #(n+60). All other advancements needed to finish a race will takes place immediately following the third one. The Draw # to start 2nd advancement, 3rd advancement will be printed on the bet ticket such as shown in FIGS. 2A and 5A. For a multirace bet the player marks to select one number for the interval of draws between two races, if, for example, 35 is selected, and if the draw number to start Race 1 is n, then the whole sequence of advancements for Race 1 takes place from Draw #n, the whole sequence of advancements for Race 2 takes place from Draw #(n+35), and the whole sequence of advancements for Race 3 takes place from Draw #(n+70). The Draw # to start Race 2 and Race 3 will be printed on bet ticket as shown in FIGS. 3A and 6A. The draw interval numbers 25, 30, 35 etc. are set under the assumption that draws will take place every half a minute so that selecting 30 for a 1race game will allow a player about 15 minutes to place makeup bets between two advancements. Selecting 35 for a multirace game will also allow about 15 minutes to place makeup bets between two races. A race ends when three racers reach their finish lines.

The random numbers of every draw will be input into the computer connected to all wagering machines to record all advancements for every race on a bet ticket. Since a racer's advancement is equivalent to the retreat of its finish spot/line, to record a race, it's practical to keep all racers in line 0 and reset their finish lines based on advancements. On request of a bet ticket holder, the computer will print out updated finish lines for a race in process as shown in FIGS. 4A and 4B, or for a whole race as shown in FIG. 4C.
Description of Placing Bets

There are 1, 2 and 3race Win/Place/Show bets using a bet slip as shown in FIGS. 2 and 5; and 1, 2 and 3race Win/Exacta/Tricta bets using a bet slip as shown in FIGS. 3 and 6. Besides, there are credit bets using hanging multirace ticket.

On a bet slip the player sets up one or more race courses as explained in ‘Description of the preferred playing surface and the race’.

On a bet slip the player marks either one ‘amount per bet’ or one ‘total bet amount’ except in the case of credit bet where ‘credit’ must be marked. All bets on a slip have the same per bet amount.

Ahead of explaining how to mark bet slip for a specific Win/Place/Show or Win/Exacta/Tricta bet it is necessary to introduce the following terminology. A bet placed by marking, for example, spot 13 in FIG. 2, involving racer i, will be referred to as a 13(i) bet. The same is true if 13 replaced by another numeral like 14, 15 or 24 and i is replaced by j or k. In the case of more than one spot being marked for a bet, such as both spot 13 involving racer i and spot 14 involving racer j are marked, it will be referred to as a 13(i)14(j) bet as well as a 13(i) bet or 14(j) bet Besides, a prime will be put behind the numeral to denote that it is a credit bet.

To place 1race Win/Place/Show bets: The player marks to select one or several spots 13, 14, and/or 15. Each selection counts one bet. A Win bet on racer i is a 13(i) bet. A Place bet on racer j is a 14(j) bet. A Show bet on racer k is a 15(k) bet. Let #13(i), #14(j) and #15(k) denote respectively the number of 13(i), 14(j) and 15(k) selections. There are #Race1=#13(i)+#14(j)+#15(k) bets.

To place 1race Win/Exacta/Tricta bets: The player marks to select one or several spots 13, 14, and/or 15. If one spot 15 is selected, then at least one spot 14 must be selected. If one spot 14 is selected, then at least one 13 must be selected. If there are only spot 13 being selected, then all bets are Win bets. If there are only spots 13 and 14 being selected, then all bets are Exacte bets. If there are spots 13, 14 and 15 being selected, then all bets are Tricta bets. In the case of Win bets, every selected racer i in spot 13 counts a bet. It is a 13(i) bet, which wins if racer i finishes first. In the case of Exacta bets, every selected racer i in spot 13 and every selected racer j in spot 14 with i≠j will be combined to form a 13(i)14(j) bet, which wins if racer i finishes first and racer j finishes second. If the player wants only specific combinations of selected spots 13 with 14 instead of all possible, then it is necessary to use separate slips. For example, using one bet slip, you can bet racers 1 or 2 finishing first and racers 3 or 4 finishing second, This is four bets. If you just want one bet on racer 1 finishing first and racer 3 finishing second and another bet on racer 2 finishing first and racer 4 finishing second, then you need to use two bet slips to place them separately. In the case of Tricta bets, every selected racer i in spot 13 and every selected racer j in spot 14 and every selected racer k in spot 15 with i≠j≠k≠i will be combined to form a 13(i)14(j)15(k) bet.

Every marked 1race bet slip will be examined and approved by a wagering machine in order to issue a bet ticket as shown in FIGS. 2A and 3A. The bet ticket shows selected finish line/spots, per bet amount, total number of bets, total bet amounts and when the race starts, and the second and third advancements take place. All selected racers will be marked by “X”.

To place 2race Win/Place/Show bets: The player does first just as placing 1race bets; then marks to select one or several spots 23, 24, and/or 25. Every selection extends all bets placed in Race 1. If the player wants only specific bets on racers i, j, k to be extended in specific ways instead of all possible, then it is necessary to use separate slips. Selected i′ in spot 23 will form WinWin 13(i)23(i′) bets, PlaceWin 4(j)23(i′) bets, and/or ShowWin 5(k)23(i′) bets. Selected racer j′ in spot 24 will form WinPlace 13(i)24(j′) bets, PlacePlace 14(j)24(j′) bets, and/or ShowPlace 15(k)24(j′) bets. Selected racer k′ in spot 25 will form WinShow 13(i)25(k′) bets, PlaceShow 14(j)25(k′) bets, and/or ShowShow 15(k)25(k′) bets. Let #23(i′), #24(j′) and #25(k′) denote respectively the number of 23(i′), 24(j′) and 25(k′) selections. Let #Race2=#23(i′)+#24(j′)+#25(k′). There are #Race1*#Race2 bets. Note that * is a multiplication sign.

To place 3race Win/Place/Show bets: The player does first just as placing 2race bets; then marks to select one or several spots 33, 34, and/or 35. Every selection extends all bets placed in Races 1 and 2. If the player wants only specific bets on racers i, j, k, i′, j′, k′ to be extended in specific ways instead of all possible, then it is necessary to use separate slips. Selected racer i″ in spot 33 will form WinWinWin 13(i)23(i′)33(i″) bets, PlaceWinWin 14(j)23(i′)33(i″) bets, ShowWinWin 15(k)23(i′)33(i″) bets, WinPlaceWin 13(i)24(j′)33(i″) bets, PlacePlaceWin 14(j)24(j′)33(i″) bets, ShowPlaceWin 15(k)24(j′)33(i″) bets, WinShowWin 13(i)25(k′)33(i″) bets, PlaceShowWin 14(j)25(k′)33(i″) bets, ShowShowWin 15(k)25(k′)33(i″) bets. Selected racer j″ in spot 34 will form WinWinPlace 13(i)23(i′)34(j″) bets, PlaceWinPlace 14(j)23(i′)34(j″) bets, ShowWinPlace 15(k)23(i′)34(i″) bets, WinPlacePlace 13(i)24(j′)34(j″) bets, PlacePlacePlace 14(j)24(j′)34(j″) bets, ShowPlacePlace 15(k)24(j′)34(j″) bets, WinShowPlace 13(i) 25(k′)34(j″) bets, PlaceShowPlace 14(j)25(k′)34(j″) bets, ShowShowPlace 15(k)25(k′)34(j″) bets. ShowWinShow 15(k)23(i′)35(k″) bets, WinPlaceShow 13(i)24(j′)35(k″) bets, PlacePlaceShow 14(j)24(j′)35(k″) bets, ShowPlaceShow 15(k)24(j′)35(j″) bets, WinShowShow 13(i)25(k′)35(k″) bets, PlaceShowShow 14(j)25(k′)35(k″) bets, ShowShowShow 15(k)25(k′)35(k″) bets. Let #33(i″), #34(j″) and #35(k″) denote respectively the number of 33(i″), 34(j″) and 35(k′) selections. Let #Race3=#33(i″)+#34(j″)+#35(k″). There are #Race1*#Race2*#Race3 bets.

To place 2race Win/Exacta/Tricta bets, the player does first just as placing 1race bets; then marks to select spots 23, 24, and/or 25. If one spot 25 is selected, then at least one spot 24 must be selected. If one spot 24 is selected then at least one spot 23 must be selected. Every selection extends all bets placed in Race 1. If the player wants only specific Race 1 bets to be extended in specific ways instead of all possible, then it is necessary to use separate slips. In the case of only spots 23 being selected, every selected i′ in spot 23 will form WinWin 13(i)23(i′) bets, ExactaWin 13(i)14(j)23(i′) bets, and/or TrictaWin 13(i)14(j)15(k)23(i′) bets. In the case of only spots 23 and 24 being selected, every selected racer i′ in spot 23 and selected racer j′ in spot 24 where i′≠j′ will form WinExacta 13(i)23(i′)24(j′) bets, ExactaExacta 13(i)14(j)23(i′)24(j′) bets, and/or TrictaExacta 13(i)14(j)15(k)23(i′)24(j′) bets. If the player wants only specific combinations of selected racer i′ with racer j′ instead of all possible, then it is necessary to use separate slips. In the case of spot 25 being selected, every selected racer i′ in spot 23 and selected racer j′ in spot 24 and selected racer k′ in spot 25 where i′≠j′≠k′≠i′ will form WinTricta 13(i)23(i′)24(j′)25(k′) bets, ExactaTricta 13(i)14(j)23(i′)24(j′)25(k′) bets, and/or TrictaTricta 13(i)14(j)15(k)23(i′)24(j′)25(k′) bets. If the player wants only specific combinations of selected racer i′ with racer j′ and racer k′ instead of all possible, then it is necessary to use separate slips. Regardless of bet type, let #Race1 and #Race2 denote the number of bets in Race 1 and Race 2 respectively. The total number of 2Race bets is #Race1*#Race2.

To place 3race Win/Exacta/Tricta bets: The player does first just as placing 2race bets; then marks to select spots 33, 34, and/or 35. If one spot 35 is selected, then at least one spot 34 must be selected. If one spot 34 is selected, then at least one spot 33 must be selected. Every selection extends all bets placed in Races 1 and 2. If the player wants only specific Races 1 and 2 bets to be extended in specific ways instead of all possible, then it is necessary to use separate slips. In the case of only spot 33 being selected, every selected racer i″ in spot 33 will form WinWinWin 13(i)23(i′)33(i″) bets, ExactaWinWin 13(i)14(j)23(i′)33(i″) bets, TrictaWinWin 13(i)14(j)15(k)23(i′)33(i″) bets, WinExactaWin 13(i)23(i′)24(j′)33(i″) bets, ExactaExactaWin 13(i)14(j)23(i′)24(j′)33(i″) bets, TrictaExactaWin 13(i)14(j)15(k)23(i′)24(j′)33(i″) bets, WinTrictaWin 13(i)23(i′)24(j′)25(k′)33(i″) bets, ExactaTrictaWin 13(i)14(j)23(i′)24(j′)25(k′)33(i″) bets, TrictaTrictaWin 13(i)14(j)15(k)23(i′)24(i′)25(k′)33(i″) bets. In the case of only spots 33 and 34 being selected, every selected racer i″ in spot 33 and racer j″ in spot 34 where i″≠j″ will form WinWinExacta 13(i)23(i′)33(i″)34(j″) bets, ExactaWinExacta 13(i)14(j)23(i′)33(i″)34(j″) bets, TrictaWinExacta 13(i)14(j)15(k)23(i′)33(i″)34(j″) bets, WinExactaExacta 13(i)23(i′)24(j′)33(i″)34(j″) bets, ExactaExactaExacta 13(i)14(j)23(i′)24(j′)33(i″)34(j″) bets, TrictaExactaExacta 13(i)14(j)15(k)23(i′)24(j′)33(i″)34(j″) bets, WinTrictaExacta 13(i)23(i′)24(j′)25(k′)33(i″)34(j″) bets, ExactaTrictaExacta 13(i)14(j)23(i′)24(j′)25(k′)33(i″)34(j″) bets, TrictaTrictaExacta 13(i)14(j)15(k)23(i′)24(j′)25(k′)33(i″)34(j″) bets. If the player wants only specific combinations of selected racer i″ with racer j″ instead of all possible, it is necessary to use separate bet slips. In the case of spot 35 being selected, every selected racer i″ in spot 33 and racer j″ in spot 34 and racer k″ in spot 35 where i″≠j″≠k″≠i″ will form WinWinTricta 13(i)23(i′)33(i″)34(j″)35(k″) bets, ExactaWinTricta 13(i)14(j)23(i′)33(i″)34(j″)35(k″) bets, TrictaWinTricta 13(i)14(j)15(k)23(i′)33(i″)34(j″)35(k″) bets, WinExactaTricta 13(i)23(i′)24(j′)33(i″)34(j″)35(k″) bets, ExactaExactaTricta 13(i)14(j)23(i′)24(j′)33(i″)34(j″)35(k″) bets, TrictaExactaTricta 13(i)14(j)15(k)23(i′)24(j′)33(i″)34(j″)35(k″) bets, WinTrictaTricta 13(i)23(i′)24(j′)25(k′)33(i″)34(j″)35(k″) bets, ExactaTrictaTricta 13(i)14(j)23(i′)24(j′)25(k′)33(i″)34(j″)35(k″) bets, TrictaTrictaTricta 13(i)14(j)15(k)23(i′)24(j′)25(k′)33(i″)34(j″)35(k″) bets. If the player wants only spectic combinations of selected racer i″ with racer j″ and racer k″ instead of all possible, then it is necessary to use separate slips. Regardless of bet type, let #Race1, #Race2 and #Race3 denote the number of bets in Race 1, Race 2 and Race 3 respectively. The total number of 3Race bets is #Race1*#Race2*#Race3.

Every marked multirace bet slip will be examined and approved by a wagering machine in order to issue a bet ticket as shown in FIG. 5A or 6A. The bet ticket shows per bet amount, total number of bets, total bet amounts and the draw numbers at which races start. Selected racers will be marked by “X”. No ‘3rd’ will be printed (see FIG. 6A) unless the bets are of type ‘Tricta’. Neither ‘2nd’ nor ‘3rd’ will be printed if the bets are of type ‘Win’.

A Win bet becomes winner if the selected racer finishes first. A Place bet becomes winner if the selected racer finishes first or second. A Show bet becomes winner if the selected racer finishes first, second or third. An Exacta bet becomes winner if both selected racers finish first and second as in selected order. A Tricta bet becomes winner if all three selected racers finish first, second, and third as in selected order. A multirace bet wins if it contains a winner in each concerning race.

A multirace bet becomes hanging if its Race 1 bet is a winner. A 3race bet remains hanging if its Race 1 bet and Race 2 bet are each a winner. A hanging bet ticket holder has the option to give up a portion of hanging bet in exchange for credit to place socalled credit bets.

To place Race 2 credit bet after Race 1: A hanging bet ticket will be used as bet slip. The player marks to select ‘credit percentage’ and either ‘new’ slip or not. In the case of marking ‘new’, a new bet slip with desired racetrack setup and bet selections must also be submitted where ‘credit’ must be marked. The Race 2 credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. In the case of no new bet slip, the player marks new selections on the hanging bet ticket. All new and original selections will be mixed together to form any possible bets just as described above for the regular betting. Any combination other than an original one becomes a credit bet. The Race 2 credit modified by selected percentage will be evenly applied to all credit bets.

Every hanging multirace ticket used as Race 2 credit bet slip will be examined by a wagering machine so that a revised ticket as shown in FIG. 5B or 6B (note that these figures are also revised after Race 2) will be issued. The revised ticket shows original data, “X” on the selected ‘credit percentage’, and new selections marked with “+”. The first three finishers of Race 1 will be printed in gray color. In the case of using an additional new slip, a new regular bet ticket will be issued which is henceforth independent, but with some carryover data from the original one.

To place Race 3 credit bet after Race 2: A hanging bet ticket will be used as bet slip. The player marks to select ‘credit percentage’ and either ‘new’ slip or not In the case of marking ‘new’, a new bet slip with desired racetrack setup and bet selections must also be submitted where ‘credit’ must be marked. The Race 3 credit modified by selected percentage on the hanging bet ticket will be evenly applied to all bets on the new slip. In the case of no new bet slip, the player marks new selections on the hanging bet ticket. All new and original selections will be mixed together to form any possible bets just as described above for the regular betting. Any combination other than an original one becomes a credit bet. The Race 3 credit modified by selected percentage will be evenly applied to all credit bets.

Every hanging 3race ticket used as Race 3 credit bet slip will be examined by a wagering machine so that a twice revised ticket as shown in FIG. 5B or 6B will be issued. The revised ticket shows original data, “X” on the selected ‘credit percentage’, and new selections marked with “=” The first three finishers of Race 1 and Race 2 will be printed in gray color. In the case of using an additional new slip, a new regular bet ticket will be issued which is henceforth independent, but with some carryover data from the original one.
Description of Placing Bets as in U.S. Pat. No. 5,795,226

In U.S. Pat. No. 5,795,226, using the preferred embodiment, there are nine racers. Racers 1 to 3 have each 14 spaces to advance, racers 4 to 6 each 15 spaces, racers 7 to 9 each 16 spaces. There are nine races. The racer carrying the race number moves first. There are Regular betting ahead of a race, Second Chance betting after the first round of advancements, and Last Chance betting after the second round of advancements. To play the former Race i: The player sets up a race course using the former Racer i as Racer 1 with corresponding finish line, the former Racer i+1 as Racer 2 with corresponding finish line, and so on, after 9 comes 1, cyclically. On this slip the player places bets just as the former Regular ones with former Racer j being replaced by Racer j−i+1, cyclically. After Draw 1, the player sets up a second race course on a new slip using the former Racer i+1 as new Racer 1 with corresponding finish line shortened as many spaces as already advanced, and the former Racer i+2 as new Racer 2 with corresponding finish line shortened as many spaces as already advanced, and so on, cyclically. On this second slip the player places bets as the former Second Chance ones with former Racer j being replaced by Racer j−i, cyclically. After Draw 2, the player sets up a third race course on a new slip using the former Racer i+2 as new Racer 1 with corresponding finish line shortened as many spaces as already advanced, and the former Racer i+3 as new Racer 2 with corresponding finish line shortened as many spaces as already advanced, and so on, cyclically. On this third slip the player places bets as the former Last Chance ones with former Racer j being replaced by Racer j−i−1, cyclically.

When the game is played using a table, the operator determines when to generate random numbers by watching if players are still making Second/Last Chance bets. Here the draws of random numbers take place automatically according to preset timing and probably fast. It also takes more time to mark a bet slip than simply to put bets on a table. So each player must select a suitable draw interval for individual race.
Description of the NonAutomatic Game

The game proceeds as the flowchart in FIG. 1 shows. Players may start placing bet as specified in ‘Description of placing bets’. The first ordered set of nine random numbers, called Draw #1, will be generated, displayed on monitors, and input into the computer for data processing. How a draw effects a race is specified in ‘Description of the preferred playing surface and the race’. After a preset period of time, regardless of betting situation and racing progress in any bet ticket, the next draw of random numbers, called Draw #2 will be generated. Consequently, a new set of random will be displayed on monitors and input into the computer for data processing. Racers move accordingly. Similarly, Draws #3, #4, etc. will follow.

The computer's data processing includes recording the progress of every race on a bet ticket, and using ‘Probability formulae’ to calculate ‘Payoffs and credits’ for every bet as specified below. Players may cash payoff and stop playing anytime.

Unless pause or stop has been regulated and announced ahead, it will go on indefinitely The Draw # will grow accordingly. Any regulated stop of the game must allow every race in process to proceed to the end.
Description of the Automatic Version

To play the automatic game one needs either a video game machine or a personal computer equipped with madetoorder software inclusive a random number generator. The computer is connected to a pointing device or touch screen monitor so that the action ‘select’ below can be executed by means of the pointing device or finger touching. Selecting any icon/item on the display screen will either highlight it or result in a new display. Selecting a highlighted item is to cancel that selection. All figures printed on paper are supposed to be black, white and gray. Now, on a monitor they can be quite colorful.

The game starts with the display of a bet slip as shown in FIG. 2 or 5 with additional icons/items named “Alternative slip”, “Ticket”, and “Account”. But there is no draw interval selection.

Selecting “Alternative slip” will switch to a Wim/Exacta/Tricta bet slip as shown in FIG. 3 or 6 if the displayed one is Win/Place/Show, or conversely.

The player places bets on screen just as in the nonautomatic game. Then selects “Ticket” to submit. If the submitted slip is incomplete or contains error, there will be a message like ‘Incomplete! Please select per bet amount.’, requiring the player to make amendment. If the submission is approved, a bet ticket as shown in FIG. 2A, 3A, 5A or 6A with additional icons “Go back”, “Go ahead”.

Selecting “Go back” will allow the player to make changes on the submitted bet slip.

Selecting “Go ahead” will result in an official bet ticket with a ticket # and icons “Bet slip”, “Run”, “Account”, but no draw number.

Selecting “Bet slip” will display a blank one to take bet.

Selecting “Run” will cause one draw of random numbers so that if it is a 1Race ticket, all racers will advance accordingly, if it is a multirace ticket, all racers of the foremost unfinished race will advance accordingly. However, the advancements will be shown by a new set of finish spots as specified in ‘Description of the preferred playing surface and the race’.

Selecting “Account” will result in a display as shown in FIG. 8. It shows the available balance, and all betting activities since the opening of that account.

Here the player can select “Ticket # so and so” to view that ticket and to run the race on that ticket or to place credit bets just as in the nonautomatic game.

Selecting “Run” on Account Statement will cause one draw of random numbers so that racers on every existing 1race ticket as well as racers of the foremost unfinished race on any existing multirace ticket will advance accordingly. The computer will update all finish spots.

“Bet slip” is to require a new bet slip while “Exit” to end the game.
Probability Formulae

Let g, h, i, j, k, l, m, n, s, u, x, y, z be natural numbers. We call the action of generating a random number from 1 to 6 ‘roll’.

A (n,s) sequence is a sequence of n rolled numbers whose sum is s. For example, (2,5)sequences are 1 4, 2 3, 3 2 or 4 1; (3,10) sequences are 1 3 6, 1 4 5, 1 5 4, 1 6 3, 2 2 6, 2 3 5, 2 4 4, 2 5 3, 2 6 2, . . . 6 1 3, 6 2 2 or 6 3 1.

Let R(n,s) denote the total number of all possible (n,s)sequences.

Obviously, R(1,1)=R(1,2)= . . . =R(1,6)=1 and R(1,s)=0 for s>6

In exactly 2 rolls we have s=2 by 1 1 only, and s=12 by 6 6 only; thus,

R(2,2)=R(2,12)=1

In exactly 2 rolls we have s=3 by 1 2 or 2 1, s=11 by 5 6 or 6 5, thus

R(2,3)=R(2,11)=2

In exactly 2 rolls we have s=4 by 1 3 or 2 2 or 3 1, s=10 by 4 6, 5 5 or 6 4, thus

R(2,4)=R(2,10)=3

Similarly we have

R(2,5)=R(2,9)=4

R(2,6)=R(2,8)=5

R(2,7)=6

R(2,0)=R(2,s)=0 for s>12

For n>2 we derive a recursion formula as follows:

Every (n,s)sequence is a onemoreroll extension of a (n−1,k)sequence where k is between s−6 and s−1. Thus,

R(n,s)=R(n−1,s−1)+R(n−1,s−2)+R(n−1,s−3)+ . . . +R(n−1,s−6)

Replacing s by s−1 we have

R(n,s−1)=R(n−1,s−2)+R(n−1,s−3)+R(n−1,s−4)+ . . . +R(n−1,s−7)

Together

R(n,s)=R(n,s−1)+R(n−1,s−1)−R(n−1,s−7), a recursion formula.

Note that R(n,s)=0 for s<n or 6n<s.

By the above formula we get all R(n,s) one by one as follows:

R(3,3)=0+R(2,2)−0=0+1−0=1

R(3,4)=R(3,3)+R(2,3)−0=1+2−0=3

R(3,5)=R(3,4)+R(2,4)−0=3+3−0=6

. . .

R(3,10)=R(3,9)+R(2,9)−R(2,3)=25+4−2=27

. . .

R(3,18)=R(3,17)+R(2,17)−R(2,11)=3+0−2=1

R(4,4)=0+R(3,3)−0=0+1−0=1

R(4,5)=R(4,4)+R(3,4)−0=1+3−0=4

. . .

R(4,12)=R(4,11)+R(3,11)−R(3,5)=104+27−6=125

. . .

R(4.24)=R(4.23)+R(3.23)−R(3.17)=4+0−3=1

R(5,5)=0+R(4,4)−0=0+1−0=1

R(5,6)=R(5,5)+R(4,5)−0=1+4−0=5

A standby (n,s)sequence is a sequence of n rolled numbers whose sum is between s−5 and s.

Examples: A standby (3,9)sequence is x y z such that 4<=x+y+z<=9. It can be 1 1 2, 1 1 3, . . . , 1 1 6, 1 2 1, . . . , 1 2 6, . . . , 1 6 2, . . . , 2 1 1, . . . , 2 6 1, . . . , 3 1 1, . . . , 3 5 1, 4 1 1, . . . , 4 4 1, 5 1 1, . . . , 5 3 1, 6 1 1, 6 1 2, 6 2 1.

A winning (n,s)sequence is a sequence of n rolled numbers whose sum is between s and s+5 . . .

Examples: A winning (4,14)sequence is x y z u such that 14<=x+y+z+u<=19. It can be 1 1 6 6, 1 2 5 6, 1 2 6 5, 1 2 6 6, 1 3 4 6, . . . 1 3 6 6, 1 4 3 6, . . . 1 4 6 6, 1 5 2 6 . . . 1 5 6 6 . . . 1 6 1 6, . . . 1 6 6 6, 2 1 5 6, 2 6 5 6, 3 1 4 6, . . . 3 6 4 6, 4 1 3 6, . . . 4 6 3 4, 5 1 2 6, . . . , 5 6 2 6, 6 1 1 6, . . . 6 6 1 6.

Every roll extends a sequence. By a onemoreroll extension, all 6 standby (n−1,s−1)sequences will result in a winning (n,s)sequence; but only 5 standby (n−1,s−2)sequences, 4 standby (n−1,s−3)sequences, 3 standby (n−1,s−4)sequences, 2 standby (n−1,s−5)sequences, and just one standby (n−1,s−6)sequence will result in winning (n,s)sequences. Thus, the probability of any standby sequence to result in a winning (n,s)sequence, denoted P(n,s), is 6*R(n−1,s−1)+5*R(n−1,s−2)+4* . . . +3* . . . +2* . . . +R(n−1,s−6) divided by 6*[R(n−1,s−1)+R(n−1,s−2)+ . . . + . . . + . . . +R(n−1,s−6)].

Consider now the case of playing alone, roll by roll, to get a winning (n,s)sequence. Let ch(n,s) be the chance function stating your chance to enter the nth roll. W(n,s)=ch(n,s)*P(n,s) will be the probability to win exactly in the nth roll. Roll 1: ch(1,s)=1, W(1,s)=ch(1,s)*P(1,s). Roll 2: ch(2,s)=ch(1,s)−W(1,s), W(2,s)=ch(2,s)*P(2,s)=[ch(1,s)−W(1,s)]*P(2,s). Roll 3: ch(3,s)=ch(1,s)−W(1,s)−W(2,s), etc.

Example: The case of s=8.

Roll 1: ch(1,8)=1.

P(1,8)=0.

W(1,8)=ch(1,8)*P(1,8)=0.

Roll 2: ch(2,8)=ch(1,8)−W(1,8)=1. P(2,8)=[6*R(1,7)+5*R(1,6)+4* . . . +3* . . . +2*R(1,3)+R(1,2)]/6*[R(1,7)+R(1,6)+ . . . +R(1,3)+R(1,2)]=15/6̂2.

W(2,8)=ch(2,8)*P(2,8)=15/6̂2.

Roll 3: ch(3,8)=ch(2,8)−W(2,8)=1−W(1,8)−W(2,8)=21/6̂2.

P(3,8)=[6*R(2,7)+5*R(2,6)+4* . . . +3* . . . +2*R(2,3)+R(2,2)]/6*[R(2,7)+R(2,6)+ . . . +R(2,3)+R(2,2)]=91/126.

W(3,8)=ch(3,8)*P(3,8)=91/6̂3.

Roll 4: ch(4,8)=ch(3,8)−W(3,8)=1−W(1,8)−W(2,8)−W(3,8)=35/6̂3.

P(4,8)=[6*R(3,7)+5*R(3,6)+4* . . . +3* . . . +2*R(3,3)+R(3,2)]/6*[R(3,7)+R(3,6)+ . . . +R(3,3)+R(3,2)]=5/6.

W(4,8)=ch(4,8)*P(4,8)=175/6̂4.

Roll 5: ch(5,8)=ch(4,8)−W(4,8)=1−W(1,8)−W(2,8)− . . . −W(4,8)=35/6̂4.

P(5,8)=[6*R(4,7)+5*R(4,6)+4* . . . +3* . . . +2*R(4,3)+R(4,2)]/6*[R(4,7)+R(4,6)+ . . . +R(4,3)+R(4,2)]=9/10.

W(5,8)=ch(5,8)*P(5,8)=189/6̂5.

Roll 6: ch(6,8)=ch(5,8)−W(5,8)=1−W(1,8)−W(2,8)−W(5,8)=21/6̂5.

P(6,8)=[6*R(5,7)+5*R(5,6)+4* . . . +3* . . . +2*R(5,3)+R(5,2)]/6*[R(5,7)+R(5,6)+ . . . +R(5,3)+R(5,2)]=17/18.

W(6,8)=ch(6,8)*P(6,8)=119/6̂6.

Roll 7: ch(7,8)=ch(6,8)−W(6,8)=1−W(1,8)−W(2,8)− . . . −W(6,8)=7/6̂6.

P(7,8)=[6*R(6,7)+5*R(6,6)+4* . . . +3* . . . +2*R(6,3)+R(6,2)]/6*[R(6,7)+R(6,6)+ . . . +R(6,3)+R(6,2)]=41/42.

W(7,8)=ch(7,8)*P(7,8)=41/6̂7.

Roll 8: ch(8,8)=ch(7,8)−W(7,8)=1−W(1,8)−W(2,8)− . . . −W(7,8)=1/6̂7.

P(8,8)=[6*R(7,7)+5*R(7,6)+4* . . . +3* . . . +2*R(7,3)+R(7,2)]/6*[R(7,7)+R(7,6)+ . . . +R(7,3)+R(7,2)]=1.

W(8,8)=ch(8,8)*P(8,8)=1/6̂7.

Since ch(8,8)−W(8,8)=1−W(1,8)−W(2,8)− . . . −W(8,8)=0, no further roll can win.

Now our race game is that every Racer i is playing with competition to get a winning (n, Tr(i))sequence where Tr(i) denotes the track length of Racer i, i.e., the number of advancement spaces from start to finish. There are P(n, Tr(i)) just as explained above. Let A(n,i) denote the probability of Racer i finishing Tr(i) first in competition in exactly n rolls.

A(n,i)=P(n,Tr(i))*ch(n,i)

where ch(n,i), the chance of reaching the nth roll, is equal to [1all A(n,i) listed ahead, if any] ordered as follows: The first n is 1 and the first i is 1. Let i run from 1 to 9, then increase n by one and run another cycle of i as before and so on till n=g with ch(n,i) being immaterial. We'll talk about some specific g later on.

The probability of Racer i finishing first is

W(i)=A(1,i)+A(2,i)+A(3,i)+ . . . +A(g,i)

Let Bm(n,i,j) denote the probability of Racer j finishing Tr(j) in competition in exactly n rolls after Racer i finishing first in exactly m rolls.

Bm(n,i,j)=P(n,Tr(i))*ch(n,i,j)

where ch(n,i,j), the chance of Racer j reaching the nth roll, is equal to [1all Bm(n,i,j) listed ahead, if any] ordered as follows: The first n is m and the first j is i+1 which is 1 if i=9. Let j run from i+1 to 9 and then from 1 to i−1, then increase n by one and run another cycle of j as before and so on till n=g.

Note that, writing computer program to calculate, one has to set Bm(n,i,j)=0 when it is so, namely

The probability of Racer i finishing first and Racer j second is

$X\ue8a0\left(i,j\right)=A\ue8a0\left(1,i\right)*\left[B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(1,i,j\right)+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(2,i,j\right)+\dots +B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(g,i,j\right)\right]+A\ue8a0\left(2,i\right)*\left[B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(2,i,j\right)+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(3,i,j\right)+\dots +B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(g,i,j\right)\right]+A\ue8a0\left(3,i\right)*\left[B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(3,i,j\right)+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(4,i,j\right)+\dots +B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(g,i,j\right)\right]+\dots +A\ue8a0\left(g,i\right)*\mathrm{Bg}\ue8a0\left(g,i,j\right)$

The probability of Racer j finishing first or second is

PB(j)=W(j)+X(1,j)+X(2,j)+X(3,j)+ . . . +X(9,j)

Let Cm(n,i,j,k) be the probability of Racer k finishing Tr(k) in competition in exactly n rolls after Racer i has finished first in m rolls or less and Racer j finishing second in exactly m rolls.

Cm(n,i,j,k)=P(n,Tr(i))*ch(n,i,j,k)

where ch(n,i,j,k), the chance of Racer k reaching the nth roll, is equal to [1all Cm(n,i,j,k) listed ahead, if any] ordered as follows: The first n is m and the first k is j+1 which is 1 if j=9. Let k run from j+1 to 9 and then from 1 to j−1, then increase n by one and run another cycle of k as before and so on till n=g.

Note that, writing computer program to calculate, one has to set Cm(n,i,j,k)=0 when it is so, namely

 for i=j or i=k or j=k
 for n=m and i<j and k<j
 for n=m and j<i
 for n=m and j=1
 for n=m+1 and k<j<i
 for n=m+1 and i=1 and k<j

The probability of Racers i, j, k finishing first, second and third respectively is

$T\ue8a0\left(i,j,k\right)=A\ue8a0\left(1,i\right)*\left\{B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(1,i,j\right)*\left[C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(1,i,j,k\right)+C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(2,i,j,k\right)+\dots +C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(g,i,j,k\right)\right]+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(2,i,j\right)*\left[C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(2,i,j,k\right)+\phantom{\rule{0.3em}{0.3ex}}\ue89eC\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(3,i,j,k\right)+\dots +C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(g,i,j,k\right)\right]+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(3,i,j\right)*\left[C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(3,i,j,k\right)+C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(4,i,j,k\right)+\dots \right]+\dots +B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e1\ue89e\left(g,i,j\right)*\mathrm{Cg}\ue8a0\left(g,i,j,k\right)\right\}+A\ue8a0\left(2,i\right)*\left\{B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(2,i,j\right)*\left[C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(2,i,j,k\right)+C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(3,i,j,k\right)+\dots +C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(g,i,j,k\right)\right]+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(3,i,\phantom{\rule{0.em}{0.ex}}\ue89ej\right)*[\phantom{\rule{0.em}{0.ex}}\ue89eC\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(3,i,j,k\right)+C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(4,i,j,k\right)+\dots +C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(g,i,j,k\right)]+B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(4,i,j\right)*\left[C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e4\ue89e\left(4,i,j,k\right)+C\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e4\ue89e\left(5,i,j,k\right)+\dots \right]+\dots +B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e2\ue89e\left(g,i,j\right)*\mathrm{Cg}\ue8a0\left(g,i,j,k\right)\right\}+A\ue8a0\left(3,i\right)*\left\{B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\left(3,i,j\right)*\left[\mathrm{C3}\ue8a0\left(3,i,j,k\right)+\dots \right]+\phantom{\rule{0.em}{0.ex}}\ue89e\dots +B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e3\ue89e\phantom{\rule{0.8em}{0.8ex}}\ue89e\dots \right\}+A\ue8a0\left(4,i\right)*\left\{B\ue89e\phantom{\rule{0.3em}{0.3ex}}\ue89e4\ue89e\left(4,i,j\right)*\dots +\dots \right\}+\phantom{\rule{0.em}{0.ex}}\ue89e\dots +A(\phantom{\rule{0.em}{0.ex}}\ue89eg,\phantom{\rule{0.em}{0.ex}}\ue89ei)*\phantom{\rule{0.em}{0.ex}}\ue89e\mathrm{Bg}(\phantom{\rule{0.em}{0.ex}}\ue89eg,\phantom{\rule{0.em}{0.ex}}\ue89ei,\phantom{\rule{0.em}{0.ex}}\ue89ej)*\phantom{\rule{0.em}{0.ex}}\ue89e\mathrm{Cg}(\phantom{\rule{0.em}{0.ex}}\ue89eg,\phantom{\rule{0.em}{0.ex}}\ue89ei,\phantom{\rule{0.em}{0.ex}}\ue89ej,\phantom{\rule{0.em}{0.ex}}\ue89ek)\ue89e\phantom{\rule{0.3em}{0.3ex}}$

The probability of Racer k finishing first, second or third is

$S\ue8a0\left(k\right)=P\ue8a0\left(k\right)+T\ue8a0\left(1,1,k\right)+T\ue8a0\left(1,2,k\right)+T\ue8a0\left(1,3,k\right)+\dots +T\ue8a0\left(1,9,k\right)+T\ue8a0\left(2,1,k\right)+T\ue8a0\left(2,2,k\right)+T\ue8a0\left(2,3,k\right)+\dots +T\ue8a0\left(2,9,k\right)+T\ue8a0\left(3,1,k\right)+\dots +\dots +\dots +T\ue8a0\left(9,8,k\right)+T\ue8a0\left(9,9,k\right)$

Note that the QBasic program File 91416 in U.S. Pat. No. 5,795,226 can be adjusted by resetting Tr(i) and other parameter values to calculate any probabilities of the above formulae. Set Tr(i)=0 to scratch Racer i. P(n,l) in U.S. Pat. No. 5,795,226 was defined slightly different from P(n,s) here, but all numerical values P(n,l) in subroutine SUB calc2P are exactly P(n,s). Because the letter l is hardly distinguishable from the number 1, it has been replaced with s here.

Let's turn to the problem of chance being immaterial under the following assumptions: 1. There are 9 racers. 2. Tr(i)<=16.3. The race ends when three racers finish. Then ch(n,i), ch(n,i,j), and ch(n,i,j,k) are all immaterial for n>=g=7 for the following reason. The summation of R(7,7) to R(7,42) is 6̂7=279936. Those of value less than 16 are: R(7,7)=1, R(7,8)=7, R(7,9)=28, R(7,10)=84, R(7,11)=210, R(7,12)=462, R(7,13)=917, R(7,14)=1667, R(7,15)=2807. Their sum is 6183. Thus, after seven rolls the chance of a racer having moved less than 16 spaces is 6183/279936<1/40. The chance of a race still having finished after 7 rolls is less than ( 1/40) ̂7 (the exponent 7 stands for racers not for rolls). This means, using g=7, the probability calculation error is less than one billionth. Using today's computer, one can neglect the ‘immaterial’ question, and let the calculation go on to g being equal to the maximal Tr(i).
Payoff, Credit and House Edge Formula

Let $a be per bet amount, e be house edge, and p1, p2, p3 be the winning probability of Races 1, 2 and 3 respectively. Let #Race1, #Race2, #Race3 be as defined in ‘Description of placing bets’, #Race2cr and #Race3cr be the numbers of Race2 and Race3 credit bets; and r2% and r3% be selected Race2 and Race3 credit percentages.

Based on ‘house edge only to be applied once at the payoff’:

Winning 1Race bet with winning probability p pays $a*(100−e)%/p.

Each of #Race2 2Race bet becoming hanging after Race 1 earns credit $a/p1 to be evenly applied to all Race 2 credit bets modified by r2%.

Winning 2Race regular bet pays $a*(100−e)%*(100−r2)%/(p1*p2).

Winning credit bet on a 2Race ticket pays $b*(10−e)%*r2%/p2 where $b is the total Race2 credit divided by #Race2cr.

Each of #Race2*#Race3 3Race bet becoming hanging after Race 1 earns credit $a/p1 to be evenly applied to all Race2 credit bets modified by r2%.

Each of #Race3 3Race regular bet remaining hanging after Race 2 earns credit $a*(100−r2)%/(p1*p2) to be evenly applied to all Race3 credit bets modified by r3%.

Each of #Race3 Race 2 credit bet on a 3Race bet ticket becoming hanging after Race 2 earns credit $b/p2 where $b is the total Race2 credit modified by r2% and divided by #Race2cr. This $b/p2 will also be evenly applied to all Race3 credit bets modified by r3%.

Winning 3Race regular bet pays $a*(100−e)%*(100−r2)%*(100−r3)0%/(p1*p2*p3).

Winning Race2 credit bet on a 3Race ticket pays $b*(100e)%*(100−r3)%/(p2*p3) where $b is the total Race2 credit modified by r2% and divided by #Race2cr.

Winning Race3 credit bet on a 3race ticket pays $c*(100−e)%/p3 where $c is the total Race3 credit modified by r3% and divided by #Race3cr.

So we just need to figure out a house edge formula that makes the game fair and attractive.

At a racetrack, experienced players check odds board before placing bets. For example, the current Win odds of horse A are 9 to 1, that of Double on A&B are x to 1, and the morning line odds of B in the next race are 9 to 1. If x=99, then it makes no difference whether to place $20 Double on A&B or to place $20 Win on A first and then to place all cashed $200 on B because the final payoff is anyway $2,000. If x is less than 99, say 94, then it's better not to place $20 on A&B because the final payoff is $1,900. If x is larger than 99, say 104, then it's better to place $20 on A&B because the final payoff is $2,100. For the racetrack operator, it's better all players simply place Win on A and then Win on B because based on, say 15% Win takeout, the total gain is $3.00 plus $30 while based on, say 25% Double takeout, only a single gain of $5.00. But Double will be offered to please horse players and many of them probably won't use all of cashed $200 to bet on B.

Now let the above wagering be nonparimutuel. ‘Odds x to 1’ becomes winning probability 1/(x+1). If both A's and B's winning probability are W(A)=W(B)= 1/10, then Double on A&B is W(AB)=1/100. Should the game operator set different house edge based type of bets, say 3% on 1Race bet, 5% on 2Race bet, 8% or 10% on 3Race bet? If so, to place $20 on A&B will result in $1900, while first $20 Win on A, then put all cashed $194 on B result in $1881.80. There is a $19.20 difference. But since the range of winning probability is not simply a matter of bet type while payoff is always operator's burden, it is more scientific to assign house edge not according to bet types but rather based on winning probability as follows:

Let house edge be e% and x=1/p where p is winning probability.

e=2+x/2 for 0<x<=5 (see FIG. 12A)

e=4.5+(x−5)/10 for 5<x<=10 (see FIG. 12A)

e=4+(n+1)*n/2+(n+1)*x/10̂(n+1) for 10<x with integer n satisfying 10*n<x<=10̂(n+1) (see FIG. 12B). Note that * is multiplication, ̂ is exponentiation.

By this formula, e=5 for x=10 while e=4+2+(100−10)/90=7 for x=100 implying n=1, the $20 bet on A&B will result in payoff $20*100*(100−7)%=$1860 while first $20 Win on A, then puts all cashed $20*10*(100−5)%=$190 on B will result in only $190*10*(100−5)%=$1805. There is a $55 difference which makes the 2Race bet incentive. Besides, the credit also allows the hanging bet holder to have second thoughts, for example, giving up B, using 100% of $20*10=$200 credit applied evenly to C with winning probability p=W(C)=⅕ as well as to D with winning probability p=W(D)=¼. Consequently, payoff for A&C is $200*100%*(100−e(AC))%/(#Race2cr*W(A)* W(C))=$468.00 where e(AC)=4+2+2*50/100=7 while payoff for A&D is 200*100%*(100−e(AD)) %/(#Race2cr*W(A)*W(D))=$372.80 where e(AD)=4+2+2*40/100=6.8. Comparatively, if the player first bets $20 on A and then puts all cashed $20*10*(100−5) %=$190 evenly on C and D, the result will be $95*(100−e(C))%/W(C))=$453.62 for C where e(C)=4+5/2=4.5, and $95*(100−e(D)) %/W(D))=$364.80 for D where e(D)=2+4/2=4. Thus, to place first a $20 2Race bet on A&B, then use 100% credit to bet on C and D has an advantage of $468.00$453.62=$14.38 for C and $372.80−$364.80=$8.00 for D than to place $20 1Race bet on A, and then to put cashed $190 evenly on C and D. But allowing credit wagering also gives the operator a passive chance to reduce the risk of a $1,860 payoff.
More Numerical Examples

After the importance of house edge has been clarified above, let's see three cases using bet slips, but without any house edge to make the calculation less complicated.

First, turn to the 1Race W/P/S bet ticket shown in FIG. 2A. Track length 14 15 16 14 15 16 14 15 16. Winning probabilities and $1.00 payoff without house edge are shown in FIG. 9A. The player makes $10 Win bets on #3, #4, $5, #7, #8, for their chances are neither best two nor least two; $10 Place bets on #3, #4, $5, #7, for their chances are neither best two, nor least three; and $10 Show bets on #6, #7, $8, #9, for their chances are the least four. Total cost: $130.

After the first round of advancements, track length 12 13 11 13 11 13 10 11 11 and winning probabilities as shown in FIG. 9B. The player makes Second chance bets as follows: $15 Win on #2, #9, for among four unselected their chances are now in the middle; $15 Place on #1, #2, #8, for among five unselected their chances are now in the middle; $15 Show on #2, #4, for among five unselected their chances are now the least two. Total cost: $105.

After the second round of advancements, track length 9 9 5 11 10 7 8 9 8 and winning probabilities as shown in FIG. 9C. The player makes Last chance bets as follows: $20 Win on #1, for #1 has less chance than #6 and there will be a Win payoff unless #6 finishes first; $15 Place on both leftover #6 and #9, for there will be two Place payoffs regardless of any finishing order; $15 Show on #1, #5 but not on #3, for #3 has now the best chance to be on the board and has been bet on to finish first or second at the beginning. Total cost: $80.

The race proceeds as shown in FIG. 4C with finishing order: 358. Payoffs under e=0:

Regular Win on #3: $10/0.073085=$136.83,

Regular Place on #5: $10/0.179608=$55.68,

Regular Show on #8: $10/0.169411=$59.03,

Last chance Show on #5: $15/0.169760=$88.36.

Net win: $339.90$315.00=$24.90. Without any makeup bets, the player could have won $121.54. Although all second chance bets are losing and only one last chance bet is winning, the player's strategy of playing slightly against odds at the beginning with middle odds makeups later on is successful. Suppose the player makes neither second nor last chance bet and 612 is the finishing order, only Show on #6 pays $59.37 so that it will be a net loss of $71.63.

Second, turn to 3Race Win/Place/Show bet ticket as shown in FIG. 5A with Race 1 Track: 15 14 16 12 15 16 13 14 13, Race 2 Track: 13 14 15 12 14 15 14 11 13, Race 3 Track: 12 15 14 14 11 13 15 12 14. There are #Race1=9, #Race2=6, #Race3=6 selections forming 324 bets of $0.10 each. Winning probabilities and $1.00 payoff without house edge are shown in FIGS. 10 A to C.

As shown in FIG. 5B, the finishing order of Race 1 is 942. For each of #Race2*#Race3=36

13(9) bet earns credit $a/W1(9))=$0.10/0.063994=$1.56,

14(9) bet earns credit $a/P1(9))=$0.10/0.176369=$0.57,

14(4) bet earns credit $a/P1(4))=$0.10/0.482636=$0.21,

15(2) bet earns credit $a/S1(2))=$0.10/0.531571=$0.19.

Thus, total credit for Race2 is 36*($1.56+$0.57+0.21+$0.19)=$91.08 to be applied evenly to #Race2cr24, based on #Race3=6 and 4 selections shown on revised bet ticket in FIG. 5B, which will be modified by r2=40. Thus, the bet amount of each of Race2 credit bet is $b=$1.518.

The finishing order of Race 2 is 835 as shown in FIG. 5B. For each of #Race3=6

13(9)24(3) bet earns credit $a*(100−r2)%/W1(9)*P2(3))=$6.85,

14(9)24(3) bet earns credit $a*(100−r2)%/P1(9)*P2(3))=$2.49,

14(4)24(3) bet earns credit $a*(100−r2) %/P1(4)*P2(3))=$0.98,

15(2)24(3) bet earns credit $a*(100−r2)%/S1(2)*P2(3))=$0.82,

Credit 23′(8) bet earns credit $b/W2(8))=$1.518/0.175024=$8.67,

Credit 25′(5) bet earns credit $b/S2(5))=$1.518/0.235294=$6.34.

Total credit for Race3 is 6*($6.85+$2.49+$0.98+$0.82+$8.67+$6.34)=6*$26.15=$156.90 to be evenly applied to #Race3cr=24, based on 4 selections shown on revised bet ticket in FIG. 5B, which will be modified by r3=60. Thus, the bet amount of each of Race3 credit bet is $c=$23.535.

Assume that the finishing order of Race 3 is 386. No house edge payoffs:

13(9)24(3)35(3) bet pays $a*(100−r2)%*(100−r3) %/(W1(9)*P2(3)*S3(3))=$0.75,

14(9)24(3)35(3) bet pays $a*(100−r2)%*(100−r3)%/(P1(9)*P2(3)*S3(3))=$2.74,

14(4)24(3)35(3) bet pays $a*(100−r2)%*(100−r3)%/(P1(4)*P2(3)*S3(3))=$1.00,

15(2)24(3)35(3) bet pays $a*(100−r2)%*(100−r3)%/(S1(2)*P2(3)*S3(3))=$0.91

Credit 23′(8)35(3) bet pays $b*(100−r3)%/(W2(8)*S3(3))=$9.57

Credit 25′(5)35(3) bet pays $b*(100−r3)%/(S2(5)*S3(3))=$7.01

Credit 34′(3) bet pays $c/P3(3)=$113.69

Credit 34′(8) bet pays $c/P3(8)=$120.08.

Last, turn to 3Race Tricta/Exacta/Tricta bet ticket as shown in FIG. 6A with Race 1 Track: 14 16 11 10 15 12 13 15 11, Race 2 Track: 13 15 15 12 13 12 11 13 12, Race 3 Track: 16 14 15 15 15 14 14 14 13. There are #Race1=75, #Race2=28, #Race3=36 selections forming 75,600 bets of $0.01 each. Winning probabilities required for calculation below are shown in FIGS. 11A to C.

FIG. 6B shows the finishing order of Race 1: 617. Each of #Race2*#Race3=1008

13(6)14(1)15(7) bet earns credit $a/p1=$0.01/0.000096=$104.1667.

Total credit is $105,000 to be applied evenly to #Race2cr=22*36=792 credit bets modified by r2=60 so that $b=132.58*60%=$79.545.

FIG. 6B shows the finishing order of Race 2: 324. Each of #Race3=36

13(6)14(1)15(7)23(3)24(2) bet earns credit $a*(100−r2)%/*(p1*p2)=$0.01*40%/(0.000096*0.000734)=$56766.58.

Total credit is $2,043,597 to be applied evenly to #Race3cr=320 credit bets modified by r3=80 so that $c=$6,386.24*80%=$5,108.99.

There are 36+320=356 out of 504 possible outcomes producing winner. Let's see three of them.

(1) Assume finishing order 248. Credit 33′(2)34′(4)35′(8) bet pays $c/p3=$2,953,516

(2) Assume finishing order 291. Credit 33′(2)34(9)35′(1) bet pays $c/p3=$1,671,572

(3) Assume finishing order 725. 13(6)14(1)15(7)23(3)24(2)33(7)34(2)35(5) bet pays $a*(100−r2) %*(100−r3) %/(p1*p2*p3)=$0.01*0.4*0.2*/(p1*p2*p3)=$7,073,716

Due to high odds all three payoffs are over a million dollars. If no credit is applied, and 725 is Race 3 finishing order, then payoff is more than 88 million dollars.
CONCLUSION

The preferred embodiment described above provides an extremely low operation cost game to be easily run by an existing or future keno/lottery kind of operator. Its automatic version can be easily integrated into some existing casino multigame video machines.

As explained, the method presented here allows us to place bets just as in U.S. Pat. No. 5,795,226. What missing there, but available here are: 1. There is no need to set up a physical race course with racing action for all players. 2. Anyone can start an individual race anytime. 3. Each player can set up a race course with own preferred number of racers and individual finish lines. 4. Credit to place optional, free makeup bets.

Naturally an inventor is not to rule how a game operator should make profit. But it is necessary to provide payoff calculation based on winning probability and house edge. Since a nonparimutuel game's popularity depends greatly on house edge, a thoughtfully constructed house edge formula is included. It is attractive and fair that the house edge increases inversely proportional to the winning probability, and no house edge will be charged till at payoff. Sophisticated players can enjoy beating probability free of charge as long as possible while the operator is always protected by the final house edge charge.

The invention is a game of any desired probability. Nowadays there are people everywhere interested in playing probability game without real big money gambling. So a game operator can organize prize tournaments to let them each start with a fixed imaginary bankroll and try to reach the best winning result within a given period of time.

Following the step by step derivation of probability formulae one can easily set up games similar to the one precisely described above. First, rolled numbers are not necessarily 1 to 6, it can be 0 to 5 or any other positive or negative integers where negative ones mean backward motion. Second, a roll is not necessarily to generate six numbers, it can be more or less. One can similarly define (n,s)sequences, R(n,s), standby and winning sequences, and P(n,s) etc. Third, since Tr(i) is a variable, the maximal track length is not necessarily equal to 16. Fourth, the number of racers is not necessarily 9. If it is q instead of 9, then in A(n,i)=P(n,Tr(i))*ch(n,i) we let i goes from 1 to q instead of 1 to 9; —everywhere modulo 9 becomes modulo q—. One can similarly form Bm(n,i,j) and Cm(n,i,j,k) to calculate all kinds of probabilities for racers i, j, k finishing first, second and third. Besides, in the same art of forming Cm(n,i,j,k), one can form Dm(n,i,j,k,l), Em(n,i,j,k,l,h) etc. to calculate the probability of a racer finishing 4th, 5th, etc. so that betting can be socalled superfacta super high 5 etc. In that case a race will end after 4, 5 or more racers reach their finish lines. And, multi races can be 4race, 5race, etc.

In the above description there is no credit bet offered between two rounds of advancements. But in order to gain credit a player can place a multirace bet with race courses set so short that each race may end within one round of advancement.

Thus, the scope of the invention should be determined by the appended claims and their legal equivalents, rather than by examples given.