If you watch any movie that involves poker, it seems like it’s only a matter of time before a royal flush makes an appearance. This hand in poker is the ten, jack, queen, king and ace, all of the same suit. Typically the hero of the movie is dealt this hand and it is revealed in a dramatic fashion. The reason for this is that a royal flush is the highest ranked hand in the card game of poker. But multiple cinematic appearances aside, how likely is it to be dealt a royal flush? What is the probability that you would see this type of hand?
Basic Assumptions and Probability
There are a multitude of different ways that poker can be played. For our purposes we will assume that a player is dealt five cards from a standard 52 card deck. No cards are wild, and the player keeps all of the cards that are dealt to him or her.
To calculate the probability of being dealt a royal flush, we need to know two numbers:
- The total number of possible poker hands
- The total number of ways that a royal flush can be dealt.
Number of Poker Hands
Some of the techniques of combinatorics, or the study of counting, can be applied to calculate the total number of poker hands. It is important to note that the order in which the cards are dealt to us does not matter. This means that each hand is a combination of five cards from a total of 52. We use the formula for combinations and see that there are a total number of C( 52, 5 ) = 2,598,960 possible distinct hands.
A royal flush is a flush, so all cards must be of the same suit. Unlike other flushes, the value of all five cards are determined. These cards must be the ten, jack, queen, king and ace. Since there are four suits of hearts, diamonds, clubs and spades, there are only four possible royal flushes that can be dealt.
Probability of a Royal Flush
We now carry out the division and see that a royal flush is rare indeed. There is only a probability of 4/2,598,960 = 1/649,740 = 0.00015% of being dealt this hand. To put this number in perspective, if you were dealt 20 hands of poker every night of the year, in 89 years you should only expect to see one royal flush. So this hand is not as common as what the movies might make us believe.