Since a flush is just below a full house in the hierarchy of poker hands, let's see how the probabilities compare. We determined, here, that the probability of being dealt a full house is 3744/2,598,690, or about 0.00144--slightly better than 1-in-700. Also, the probability of drawing to a full house from two pair is 4/47, about 8.5%.

You can think of the probability of being dealt a flush in the following manner. You're dealt five cards. The first one can be anything. The second one has to be the same suit as the first. There are twelve of that suit left, and 51 unknown cards, for a probability of 12/51. And, if that chance is realized, you're down to 11/50 for the third card, then 10/49 on the fourth, and 9/48 on the last card. The probability of a flush is the product of the probabilities of these separate events, which comes out to 11,880/5,997,600, or very close to 0.00198, which is very close to 1-in-500. (This represents the probability of all flushes, including "straight flushes," such as the 4, 5, 6, 7 and 8 of spades.) Since a full house beats a flush (except for the very rare straight flush), you'd expect the former to be less likely, and indeed it is: out of every 3500 hands, you'd expect about seven flushes on the deal, and about five full houses.

If you have any doubts about my method of calculating the probability of a flush, you can check the result against the number of possible combinations that make a flush divided by all possible combinations. The number of different ways of having a flush is given by "13 choose 5" (the number of ways of choosing five cards from thirteen) times "4 choose 1" (since you can choose five from thirteen in any one of the four suits). The number of possible five-card hands is given by "52 choose 5." Grinding the numbers, you get 5148/2,598,960, which is the same as 11,880/5,997,600.

Remember how, starting from two pair, you have about an 8.5% chance of drawing to a full house? It raises the question: what if you are dealt four cards of the same suit? what is the chance of drawing to a flush? Well, if you have, say, four hearts and one of something else, then there are 47 unknown cards, of which nine are hearts, and 9/47 is just over 0.19--more than twice the chance of drawing to a full house from two pair. Of course, if you have two pair, and the 8.5% chance of improving to a full house fails, you still have two pair, which will beat four hearts most of the time.

## Comments