[BruceZ]

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I have a problem that keeps coming up that seems basic but I can't figure out. There are some circumstances where you account for different combinations of possible outcomes and some where you don't and I can't figure out what separates the two situations.

For example, say you have AA in holdem and you want to know the probability of flopping exactly one ace and the probability of flopping no aces.

For exactly one ace you have Axx, xAx, xxA flops where x are non ace cards, or (( 2 x 48 x 47 )/ ( 50 x 49 x 48 )) x 3 combinations.

For no aces you have (48 x 47 x 46) / (50 x 49 x 48) and don't have to consider possible combinations.

This seems pretty basic but I'm constantly coming across situations where I'm not sure if I need to account for combinations. What's the determining factor?

Thanks

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In the first case you multiply by 3 because the A can come in any of 3 positions, and 2/50 * 48/49 * 47/48 by itself assumes that the A is on the first card only. In the second case, all of the cards are non-aces, so all possible orderings are already taken into account.

You can also do the first case as 2*C(48,2)/C(50,3), and the second as C(48,3)/C(50,3). Since the denominator in each case counts flops without regard to order, the numerator does the same.

Combinations often make things easier, but some problems like this one can be done both ways, and some problems are much easier with fractions, or equivalently permutations. Often it is a good idea to do it both ways when possible to make sure you get the same answer.