What is the probability that a five card poker hand contains exactly one ace?

Several different approaches can be used to answer the question, what is the probability that a five card poker hand contains exactly one ace? The number of ways to draw a royal flush, for example, is calculated by dividing the number of ways to draw a royal flush by the number of distinct combinations of cards that are available to you.

A hand with 2 aces in a row is not impossible. In fact, the probability of doing it is approximately 50%. Likewise, the probability of choosing a three of a kind is relatively low. But, there are some five-card poker hands that are not possible with a deck of 52 cards.

There are actually five distinct ways to draw a royal flush. One way is to draw the king and queen of the suit in which the card is in. Another is to draw the king and ace in the same suit. The third way is to draw the king and jack of the same suit, and so on.

A similar number of ways to draw a royal flush is to draw the king and the queen of the same suit, or to draw the king and jack of each suit in reverse order. The odds of drawing this hand are approximately 259,739:1.

The number of ways to draw the aforementioned three of a kind is somewhat more complex. In fact, you can draw four nines in 4C3=4 ways, or three nines in 4C3=3 ways, or two nines in 4C3=2 ways.