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Bad Beat Jackpots
At what point does it become profitable to play at an online hold'em bad-beat jackpot table given that it is raking fifty cents per hand to fund the jackpot? In other words, we want to know when is the fifty cent cost per hand less than what one expects to win per hand.
What is interesting is the journey to reach the number. There is no deep mathematics involved, but there are several junctures at which assumptions must be made. We discuss these assumptions carefully and try to assess the impact of the assumptions on the number derived.
The first step is the derivation of the probability that a 10-handed hold'em game produces a bad beat jackpot given that no player folds a hand until it is impossible for that hand to qualify for a bad-beat jackpot. In other words, if a player holds a hand such as 2h-5h, the player sees the flop because the hand could develop into a straight flush. If no ace, trey, four, or six of hearts comes on the flop, the player may fold the hand. Otherwise, the player sees the turn, and so on.
We now determine the preceding probability exactly. Since we don't care which players qualify for the jackpot, we are interested in the total number of possible semi-deals in 10-handed hold'em. There are C(52,5) = 2,598,960 ways to choose the board. There are then C(47,20) = 9,762,479,679,106 ways to choose 20 cards to form the 10 hands. Given 20 cards, there are 19!! = 654,729,075 ways to divide the cards into 10 hands of two cards each. Multiplying these three numbers gives us the total number of semideals for a 10-handed hold'em game. This is a magnificently large number, with 29 digits.
All we need to do now is count the number of semideals that qualify for the bad-beat jackpot and divide it by the preceding number. Thus, we now need the rules that determine whether a hand qualifies. We are going to use the rules posted at PartyPoker. Essentially, the rules stipulate that quad 8s or better must be beaten, the player must use her best hand, and both hole cards must form her best hand. We still have one interpretive question about their rules, but unfortunately no one from PartyPoker has yet answered our query. Here is the question and our interpretation.
Suppose one player has K-10, another player has J-9 of hearts, and the board has 10-10-10-Q-K, where three of them are hearts. The player with J-9 of hearts has a straight flush, and the other player's best hand is quad 10s with a K kicker. We are going to interpret the hand with quad 10s as qualifying even though her kicker in the hole ties the kicker on theboard. The boards that allow a bad-beat jackpot may be classified as follows: full house on board; trips on board with two other cards that allow a straight flush; two-pair on board; one pair on board with three other cards that allow a straight flush; or five distinct ranks on board that allow two players to hold simultaneous straight flushes.
All we need to do now is count the number of semideals for each of the preceding types of boards that produce two or more players with qualifying hands. We shall not provide the details for this straightforward, though tedious, computation. The probability turns out to be very close to 1/160,000.
And remember this means a deal happens at your table, but it does not necessarily means you are one of the two players involved. Multiply the above number by 50 cents, and you would need the bad beat jackpot to be worth $80,000 so that it makes sense to play at that table.
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