Her geeky husband looking over her shoulder, my wife Amanda prepares for her final exam in Discrete Mathematics. More than one of the exercises my eye falls on seem to rebut the familiar complaint, "This is so impractical and will never help me in real life!" For example: What is the probability of being dealt a full house in five-card poker?

Well, that would be the ratio of the number of different ways there are to have a full house to the number of unique hands one might be dealt in five-card poker. Since there are 52 cards in the deck, the latter is the number of different ways of selecting five from 52. "Fifty-two choose five" is given by 52 factorial divided by the product of 47 factorial and 5 factorial, denoted

52!/(47!)(5!) = 52 x 51 x 50 x 49 x 48/5 x 4 x 3 x 2 x 1 = 2,598,960.

Now, how many ways are there of having a full house? In one denomination, you need three of the four cards--three of the four aces, say. There are four ways to choose three from four. You need two of the four cards in a second denomination, and there are six ways of choosing two from four (4!/2! x 2! equals 6). So the product of four and six--24--represents the number of ways there are of having a particular full house, like "tens over queens." Since there are thirteen denominations (ace through king), there are thirteen choices for one denomination and only twelve for the other--obviously, if you have three tens, ten is not a possibility for the pair. Putting this all together, we have 24 x 13 x 12 ways of having a full house: 3744.

And 3744 divided by 2,598,960 is about 0.00144, or 1-in-694.

A full house is two pair plus a good draw, so a poker player should know the probabilities along this line. Turns out you are 33 times more likely to be dealt two pair than a full house. If you are dealt two pair, the probability of drawing to a full house is given by 4 (the number of remaining cards that will improve your hand)/47 (the number of unknown cards). Four divided by 47 is 0.0851, or between 8 and 9 percent.

We geeks have a hard time understanding why more people aren't interested in this kind of stuff. Surely they don't think poker is remote from "real life"?

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