The Probability of Being Dealt a Royal Flush in Poker

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 is a poker hand that has a very specific composition: 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. A royal flush is the highest ranked hand in the card game of poker. Due to the specifications for this hand, it is very difficult to be dealt a royal flush. 

Basic Assumptions and Probability

There is 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.

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Once we know these two numbers, the probability of being dealt a royal flush is a simple calculation. All that we have to do is to divide the second number by the first number.

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. Since the order 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.

Royal Flush

A royal flush is a flush. This means that all of the cards must be of the same suit. There are a number of different kinds of flushes. Unlike most flushes, in a royal flush, the value of all five cards are completely specified. The cards in one's hand must be a ten, jack, queen, king and ace all of the same suit.

For any given suit there is only one combination of cards with these cards. Since there are four suits of hearts, diamonds, clubs, and spades, there are only four possible royal flushes that can be dealt.

The Probability of a Royal Flush

We can already tell from the numbers above that a royal flush is unlikely to be dealt. Of the nearly 2.6 million poker hands, only four of them are royal flushes. These nearly 2.6 hands are uniformly distributed. Due to the shuffling of the cards, every one of these hands is equally likely to be dealt to a player.

The probability of being dealt a royal flush is the number of royal flushes divided by the total number of poker hands. 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.

Much like very large numbers, a probability that is this small is hard to wrap your head around. One way to put this number in perspective is to ask how long it would take to go through 649,740 poker hands. If you were dealt 20 hands of poker every night of the year, then this would only amount to 7300 hands per 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.