Poker puzzle
Bill the lizard has an excellent post describing an interesting puzzle as to the most desirable full house in a game of poker. The rest of this post depends upon you having read that puzzle so you may wish to do that now, or skip this post. Update: The above link seems to be dead now, so I’ll just give my description of that puzzle. The question is, in 5-card draw, what is the best full-house to hold? The highest ranked full-house is aces-over-kings AAAKK. But this is 5-card draw, so if you have aces-over-X then no-one else can beat your hand with a full-house, even if you have aces-over-twos. So what you want to consider is hands which can beat your full-house. Those are 4-of-a-kind, and straight-flushes (including the royal flush 10-J-Q-K-A of the same suit). The pair of your full-house will prevent one 4-of-a-kind hand, regardless of what the pair is. But for straight-flushes, what the pair is affects which straight-flushes are prevented. A pair of Kings, only prevents two royal-flushes, and two straight-flushes running 9-K. In addition, since there are only 4 suits, at least one of your Kings must share a suit with one of your Aces, hence at least one of the royal-flushes is already prevented by one of your three aces. So, we can see that a pair of queens would be better as this would also prevent the same two 9-K straight-flushes as well as two 8-Q straight-flushes. Going further, a pair of Jacks would prevent the same 9-K, and 8-Q straight-flushes and additionally two 7-J straight-flushes. Continuing this logic, you can see that it’s better to actually have a pair that is outside of the royal-flush range, at least one of which is already prevented by one of your three aces. So you want AAA-XX, where X is lower than 10. Note also that an Ace also prevents A-2-3-4-5. But furthermore a 3 prevents more straight-flushes than a 2, since that only prevents A-5, and 2-6, wheres as 3 prevents A-5, 2-6, and 3-7. Similarly a 4 is better, and a 5 is better, but again, at least one of your 5s must share a suit with one of your aces, so you want a pair that is at least 6. So the best full-houses are AAA-XX where X is 6, 7, 8, or 9. ...