Poker Digest Vol. 3, No. 24, November 17 - 30, 2000
In the first article of this series dealing with bad-beat jackpots, we saw a few basic ideas about computing probabilities of bad-beat jackpot occurrences. One of these ideas is the fact that we are interested in semi-deals, and we determined the total number of semi-deals possible in a 10-handed hold'em game. We shall continue to call this number X and, in fact, presented it in all its glorious digits last time.
We worked out a simple example for hold'em and observed what happens when one loosens the restriction that both hole cards must play in order for a hand to qualify. We also worked out what tends to happen as the number of players in the game changes. We then promised a more detailed look at hold'em, so let's do it!
Let's first clarify our rules for qualification of a hand. All of the results we mention are for a 10-handed game. For a player's hand to qualify, the player must use the best hand she can make and both of her hole cards must play. Thus, a player holding 4-5 of spades with the 6-7-8-9 of spades on board does not have a qualifying hand because her best hand is a straight flush to the nine of spades which uses only one card in her hand. Finally, one last restriction is that in the case of four-of-a-kind, the player must have a pair in the hole for her hand to qualify.
I decided to directly compute the probability of bad-beat qualifying semi-deals which means counting the total number of qualifying semi-deals and then dividing by X to get the probability. Little did I realize as I started the computation just how ugly it was going to get. I realized ahead of time it was going to be a fairly complex job and would exemplify all the features of this kind of problem. What are those features?
First, the problem should be broken into smaller subproblems none of which in themselves are wildly difficult.
Second, the real problem is a bookeeping one of keeping track of the smaller subproblems, making sure everything is counted at least once, and eliminating multiple counting of the same semi-deals.
Third, one is dealing with large numbers -- increasing the chances of errors when entering them in equations and recording them.
As it has turned out, I have not had time to compute all of the probabilities I originally envisaged and, because of time pressure, I have not had time to do the kind of careful check I normally like to perform. Nevertheless, some values have emerged and I hope to fill in two missing values by the time this series ends.
Here's a brief sketch of how the problem is broken down. In the first article we saw how easy it is to count the number of semi-deals with two players having quads. It's easy is because there are severe restrictions on the board, and there is only one way to complete the board to four-of-kind via a player's hand. The first extension of this is to allow straight flushes, that is, we are computing the probability of a bad beat qualifying semi-deal for any four-of-a-kind or better being beaten. This still isn't so bad because even though the restrictions on the board are not as severe, the number of ways the boards can be completed to qualifying hands is easy to determine. It turns out the probability of a semi-deal occurring in which four-of-a-kind or better loses is .00001067. This is essentially one in a 100,000.
Most cardrooms with bad-beat jackpots have the minimum hand as aces full of something. Frequently, the something seems to be 10s or jacks. Once we have to start considering aces-full, the number of subcases proliferates incredibly. You must consider many different patterns involving aces on the board, and you must consider whether aces-full lose to another hand with aces-full, or lose to another hand with either quads or a straight flush. When aces-full loses to another aces-full, we must use inclusion-exclusion arguments to get exact values for many subcases. This is very time consuming.
I shall be posting full details of the computation on my web site, but I will be surprised if anyone gets through it. I already have 28 pages of output and there are several places where details are omitted. Aces-full-of-jacks and aces-full-of-queens have yet to be included as well.
Here are the numbers I know so far. The probability of a bad beat qualifying semi-deal when aces-full-of-10s must be beaten is .00004803, when aces-full-of-kings must be beaten is .00001452, and when any four-of-a-kind must be beaten is .00001067.
In rough terms this is saying the probability of four-of-a-kind or better being beaten is 1-in-a-100,000; the probability of aces-full-of-kings or better being beaten is 1 in 69,000; and the probability of aces-full-of-10s or better being beaten is 1 in 21,000. A hold'em table that is open 24 hours a day every day and averages 30 hands an hour generates more than 21,000 hands a month. This gives you an idea of the scales involved in the above probabilities. We shall say more later about the frequency of bad-beat jackpots.
As a final observation, note that the above probabilities are for the occurrences of qualifying semi-deals. Not all semi-deals are played out to the end; that is, some players who would have eventually had a qualifying hand may fold early in the action. Thus, the probability of a bad-beat jackpot actually occurring is even smaller than the numbers determined above. How much smaller is impossible to measure accurately as it depends on the looseness of the game.