For a problem as complicated as this, I would suggest a Monte Carlo model, i.e. simulation. For each card, simulate a large number of games and tote up the outcomes.
Hi, I am a math developer for a gaming technology and would like to discuss the following problem:
We have a standard bingo card, with five rows, five columns, and a free space in the center. Column 1 contains 5 integers chosen at random from the set 1 thru 15. Column 2 ... 16-30. Column 3 contains 4 integers ... 31 thru 45. Column 4 contains 5 integers ... 46 thru 60. Column 5 ... 61 thru 75. This game is a slot machine version of bingo, where only one bingo card can used per game, and every card eventually achieves a bingo.
The Draw # is defined as the number of balls drawn when the player FIRST arrives at bingo. For instance if a player hits bingo after 4 numbers have been called, then the Draw # is 4.
If a number is called such that the player has this number on their card, then we call this a daub. Important note: A player can have only so many daubs before a bingo is reached.
The Pay Group # is defined to be the ones digit of the integer that results when the four "corner" numbers are summed together. For instance, if the four corner numbers are 6,12, 68, 71, then the pay group is 7. Important note: Column 1 can only contain integers contained in the set 1 thru 15, Column 5 ... 61 thru 75.
The pay table used to calculate winnings, if any, is based on three factors. Factor #1) The Draw #, Factor #2) The number of daubs, Factor #3) The Pay Group #. For instance if the Draw # is 9, number of daubs is 4, and the paygroup is 7, then the player receives a 2 to 1 return on their bet. Important Note- The paytable details are not important to this problem, the 3 factors used in determining pay are important.
Heres the problem: I need to calculate the probability of the occurence of each winning hand. The probabilites of the losing hands are not as important. Now, which events are winners has already been determined, and I can supply the list of winning events, if necessary. For instance, I need to calculate the probabilities of the events,
5 draws, 4 daubs, Paygroup: 0
7 draws, 6 daubs, Paygroup: 0
11 draws, 9 daubs, Paygroup: 0
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Any ideas on how to solve this?