How to calculate the probability that one poker hand would beat another

Question

I would like to know how one would go about calculating the probability that any one hand of poker is any better than that of one's opponent(s).

Answer 1

There can be several interpretations for your question.

In any variant of poker, the aim is to constitute a hand of five cards, making the best combination possible. These combinations are in the order: nothing, one pair, two pairs, a set, a straight, a flush, a full house, quads, a straight flush.

In order to understand why the order is the given one, one has to compute the number of hands giving some combination, among all possible hands of five cards. For instance, a straight flush is determined by its flush and the lowest card which must be between A (= 1) and 10. This gives $10*4 = 40$ combinations. For quads, you decide the value of the quads ($13$ possibilities) and then the fifth card ($48$ possibilites). This gives $13*48$ combinations of quads.

One can do a similar analysis for any combination and you find that the order respects the probabilites of these combinations. The wikipedia article http://en.wikipedia.org/wiki/List_of_poker_hands precises these computations.

Another way to interpret your question is more practical. The most played variant for at least $10$ years is the Texas Hold'em. In this variant, you have two cards, unknown by your opponents, and five other cards will be dealt and can be used by all opponents. And you have to make the best combination of $5$ cards among the $7$ cards at your disposal.

In this variant, it is crucial to know the probability that your two cards will beat the two opponent cards when the five other cards will be dealt. For instance, a pair of aces will win about 83% of the time against a pair of kings. Of course, this doesn't take into account the diffent actions of the opponents when the successive cards are dealt.

A list of the standard situations can be found here: http://www.pokerology.com/lessons/math-and-probability/.

You can go on and ask the probability of winning the hand at any time of the hand, knowing your two cards, your two cards opponent and the common cards already dealt. Such a calculator can be found here: http://www.cardplayer.com/poker-tools/odds-calculator/texas-holdem.

Any serious poker player has to learn some probabilities (but not too much, at least in Texas Hold'em) in order to make good decisions but, of course, this is not sufficient to be a good player.