Poker Digest Vol. 4, No. 1, December 29 - January 11, 2001
Since starting this series, I have invested considerable time on these bad-beat jackpot calculations. There have been some refinements and corrections of what appeared earlier. In particular, the earlier article on Omaha bad-beat jackpot probabilities contained an unbelievable typo which I noticed immediately upon seeing the final article. I wrote C(47,20) instead of C(47,40) and then talked about the trickiness of working out the number of semi-deals for a fixed set of 20 cards. It certainly would be tricky to deal 40 cards with only 20 available. I then looked at my web site where the detailed calculations for Omaha are posted and found that I used the correct numbers there. Since I wrote the article based on my web site calculations, I do not understand how the 20s replaced the 40s. At least the probabilities are correct.
Following are three tables with bad-beat probabilities. The meaning of minimum qualifier is that in order for a bad-beat jackpot to occur, there must be two or more players whose hands are at least as good as the minimum qualifier, and one of the players must beat the other player. Two further assumptions for hold'em are that a player must use her best hand and both hole cards must play, and an assumption for both hold'em and Omaha is that in order for quads to qualify, the player must have a pair of that rank in her hand. The listed probability is the probability of a semi-deal occurring which produces a bad beat satisfying the minimum qualifier condition.
|aces full of kings||.00001394103662|
|aces full of queens||.00002358725320|
|aces full of jacks||.00004115079053|
|aces full of tens||.00006192090428|
Now let's discuss the probabilities worked out above. They all arise from exact computations, but because of the massive enumeration trees involved in the computations, all of them represent an approximation at some stage.
Many of the subcases in Omaha, for example, were enumerated completely. The seven-card stud numbers result from the fuzziest approximations. If there are no errors, then the hold'em and Omaha numbers are accurate to about five significant figures, whereas, the seven card stud numbers are accurate to about three significant figures.
Speaking of errors, in this kind of computation errors are always possible. There are sufficiently many cases that it is possible to simply omit a case, the numbers are large enough that transcription errors may creep in, and there are numerous uses of inclusion-exclusion which makes errors more likely. I intend to refine these computations over time and invite anyone who is interested to poke around the more detailed versions on my web site.
Another important fact to keep in mind is that these numbers refer to semi-deals. Not all semi-deals are played to the end, and this creates a real distortion in the probabilities. Just how much the numbers are distorted by game conditions seems like a hopeless problem to me. If you are playing in a hold'em game where people see the flop with any pair, then many of the semi-deals containing quads will come to fruition. The same can be said about straight flushes if many of the players in your game play any two suited cards with small gaps. The seven-card stud numbers have an additional problem since they were derived for semi-deals with seven players. The game is usually dealt to eight players and, as we saw in part I, this tends to increase the chances of a bad-beat jackpot qualifier. On the other hand, since hands develop slower in seven-card stud, I believe a fair number of bad-beat semi-deals will not develop because players will have folded the cards immediately.
When I told a local player about these calculations, his reaction was that he could find them easily by just running many simulations. One has to be careful with this approach. We are talking about events with very low probabilities. First of all, one must run the simulation an immense number of times. For example, the probability of a bad beat qualifying semi-deal in hold'em with four deuces as the minimum qualifier is just about spot on at one in a hundred thousand. Even 10,000,000 trials gives you only about a seventy percent chance of being accurate within ten percent of the true expectation of 100 successes. Another problem with simulations is that very low probability events may be badly measured because of a poor quality random number generator.