Probabilities of poker
In five-card draw poken, players are trying to obtain the best possible combination to win. There are \({52}\choose{5}\) possible hands:
One fine hand to end with, is four-of-a-kind, e.g., four aces, or four sevens, etc. How many ways are there to get such a hand? The hand is build in successive steps, so we can use the multiplication principle in our calculations.
First there is the rank choice. There are 13 ranks in a playing card deck and we chose one. Formally, we can write \({{13}\choose{1}}\) but this evaluates to \(13!/(12!1!)=13\). The second step involves the suit. There are four suits and our choices are \({{4}\choose{1}}=4\). And finally, we have the last step: the fifth card that can be anything else (except for the rank that we use for the four-of-a-kind). There are \(12\) such choices. All together, we have \(13\cdot 12\cdot 4=624\) ways to pull a four-of-a-kind hand.
And so we can write:
Let’s look at another hand, the single pair: two of the five cards are of the same rank. The successive steps, and their corresponding factors in the multiplication principle, are:
There are, in other words, 1,098,240 different ways to end with a single pair. And the probability is