Kevin's Math Page

 Kevin Gong January 8th, 2007

In this analysis, I'll exam bad beats in Texas Hold'em poker. For those of you not familiar with this form of poker (who have perhaps been hiding under a rock the last couple years), please reference other web pages to read about the rules. I'll assume you know the basics in this article.

For the purpose of this analysis, a bad beat is a situation in which you receive an unexpectedly bad outcome. How unexpected? Let's look at what would generally be conceived as the worst possible pre-flop bad beat. I wrote a computer program to calculate different hands against each other, and the worst case is KK vs K2o, where one of the kings is the same suit as the 2. For example, KH KD vs. KC 2H. In this case, the pocket kings win 94.92 percent of the time. Put another way, the pocket kings will lose one out of every 19.7 times. To lose this would be the worst possible pre-flop bad beat in terms of expected value -- you had 94.92% pot equity, and received 0.

 The worst possible pre-flop bad beat in terms of expected value is KK vs. K2o. The K2o wins 1 out of every 19.7 times.

As an aside, the worst hand to have against AA is A9o, which wins 1 out of every 16.9 times. The best hand to have against pocket aces is 65s, which will win 1 out of every 4.34 times. Quite a difference! (no, I am not advocating treating 65s like a great hand, but against pocket aces it is the best)

Oh, but wait...there's one hand we didn't discuss. What about AA vs. AA? Sure, it's unlikely, but in this case, how likely are you to lose? Since you basically have the same hand, you can only lose to a flush. What are the chances of 4 or 5 hearts coming on the board? Note that it's possible that 5 hearts come and make a straight flush that splits the pot: 23456, 34567, 45678, 56789, 6789T, 789TJ, 89TJQ, so we have to take those into account.

• 5 hearts (no split): ((12 choose 5) - 7) / (48 choose 5) = 785 / 1712304 [because there are 12 hearts left, and 48 cards total out of the original 52 card deck; there are 7 hands which split the pot]
• 4 hearts: (12 choose 4) * 36 / (48 choose 5) = 45 / 4324 [there are 36 non-hearts left in the deck]
• total: (785 / 1712304) + (45 / 4324) = 18605 / 1712304 = approximately 1 / 92.03

Since your opponent can hit either of two different suits, the probability of him winning is 2 * (1/92.03) = 1/46.02. This means that you will lose the entire pot only once out of 46.02 times. That's much worse than 1 out of 19.7 times. In terms of expected value, however, this is definitely not close to the KK vs. K2o case, as your expected value is 50% of the pot. So getting 0 after expecting 50% in some sense is not a horrible beat.

Still, there are hypothetical situations in which we would care more about not losing, rather than our expected value. For example, suppose you are on the bubble in a tournament (i.e., the next person out gets no prize money, but everyone after that does). You only have enough money to cover the big blind, but there are many other players left who have even less than you do. So all you want to do is survive this hand and make it into the money. If everyone folds to the small blind, and you look down and find pocket aces in the big blind, you actually would love it if your opponent had AA, as that would mean you will survive 45/46 times (97.8%). On the other hand, if your opponent had A9o, you would only survive 94.08% of the time.

The question is, is losing with AA vs. AA the worst possible pre-flop bad beat? Nope. Let's examine the case of AKo vs. AKo, where you both have the same suits. For example, AH KD vs. AD KH. In this case, you can only lose with the AH KD if 4 or more diamonds come (but not the following 5-diamond cases: 23456, 34567, 45678, 56789, 6789T, 789TJ, x9TJQ, or the 4-diamond case QJT9x), or if it comes QJT9 of hearts and you lose to a straight flush. Any other heart flush is fine, since you have the AH:

• 5 diamonds (no split): ((11 choose 5) - 13)/ (48 choose 5) = 449 / 1712304 [because there are 11 hearts left, and 48 cards total out of the original 52 card deck, and 13 cases which give you a straight flush to at least split]
• 4 diamonds: ((11 choose 4) - 1) * 37 / (48 choose 5) = 259 / 36432 [there are 37 non-hearts left in the deck; remove the one case of QJT9x]
• QJT9 of hearts: 44 / (48 choose 5) = 1 / 38916
• total: (449 / 1712304) + (259 / 36432) + (1 / 38916) = 2111 / 285384 = approximately 1 / 135.2

Much worse than 1 out of 46. But is it the worst? No, in fact the worst case is 32o vs. 32o. Let's take the example of 3H 2D vs. 3D 2H. You lose with the 3H 2D if the board shows 4567x diamonds (but not 45678), or if 4 diamonds come:

• 4567x diamonds (not 45678): 6 / (48 choose 5) = 1 / 285384
• 4 diamonds: (11 choose 4) * 37 / (48 choose 5) = 185 / 25944 [there are 37 non-hearts left in the deck]
• total: (1 / 285384) + (185 / 25944) = 509 / 71346 = approximately 1 / 140.17

 The worst possible pre-flop bad beat in terms of least likely to lose is 32o vs. 32o where the suits match. You opponent will win 1 out of every 140.07 times.

So there you have it. I am sure you will become a much better poker player armed with this knowledge. Just kidding!

Kevin's Math Page