Poker: Finding Expected Values of High Hands

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  • 0:04 Probability in Poker
  • 1:56 Probability of a Royal Flush
  • 3:09 Probabilities of…
  • 8:47 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

In this lesson, you will learn the statistical probability of the high hands in poker, including the royal flush, straight flush, straight, and flush! This lesson will also cover the combination formula.

Probability in Poker

Caleb is playing a poker game with his buddies, and he's having a lot of luck. After winning a hand with a flush, Caleb begins bidding high again on the next hand. His friends begin to wonder, what is the probability that Caleb has another great hand? To figure out the probability of a poker hand, you will need to understand how to use a combination formula, which is a probability formula that uses factorials to find the number of possible combinations of all the outcomes in the experiment.

The combination formula looks like this:

Combination formula
combination formula

You may notice that this formula uses an exclamation point, also known as a factorial in mathematics. You will need to use a graphing calculator or try a search on the Internet for 10! to find the factorial values.

Before we discuss the combination formula and how to solve poker probabilities, let's first discuss the characteristics of a fairly shuffled deck of playing cards. A fair deck of playing cards has 52 cards, not counting the jokers, and has been thoroughly shuffled. We're assuming no one has been cheating and putting extra cards in to the deck or putting the deck in a particular order.

You should also know that there are 4 suits of cards: hearts, spades, clubs, and diamonds. And there are 13 different cards in each suit: Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King. If we were to calculate the probability of drawing a single card from the deck, let's say a 10 of diamonds, then that would be 1 out of 52 chance, considering that there is only one 10 of diamonds in the deck. If we were calculating the probability of drawing an ace out of the deck, then that probability would be 4/52, because there is one ace for each suit and a total of 52 cards.

Now that we've discussed the characteristics of a deck of playing cards, let's go over the probability of each type of poker hand.

Probability of a Royal Flush

A royal flush is made up of the Ace, King, Queen, Jack, and 10 of all the same suit. Since there are four suits, then that means there are four different royal flushes possible in the deck. But we aren't done with figuring out the probability of a royal flush just yet. We need to figure out all of the possible combinations of royal flushes in comparison to all of the other possible hands you could get in a deck when dealt five cards.

First, let's look at all of the possible combinations you can get when dealt five cards. That's a combination of 52 choose 5. Take a look at our combination formula:

combination formula 52 choose 5

The result is a total of 2,598,960 combinations. Wow, that's a lot of combinations! Now we need to determine the possibility of getting a royal flush. Remember, there are four different royal flushes possible in the deck, so we can divide 4 by 2,598,960 to get our probability. That's the total number of possible royal flushes divided by the total number of possible hands. Which gives us 4/2,598,960, or a 1 in 649,740 chance! This means that Caleb has a pretty small chance of having a royal flush.

Probability of Straights and Flushes

Time for a little more poker terminology. A straight flush is another type of poker hand that is all of the same suit. Unlike the royal flush however, a straight flush just has to be 5 consecutive cards, such as a 6, 7, 8, 9, and 10 or a Queen, Jack, 10, 9, and 8. A straight flush can be any combination of cards as long as they are the same suit and are consecutive. When finding the probability of straight flushes, we do not count a royal flush, even though technically a royal flush is also a straight flush.

The easiest way to solve this particular problem is to examine how many hands you could have by looking at the highest card in your hand. For example, if you start your highest card being a king in a straight flush, the cards that follow would have to be Queen, Jack, 10, and 9. So what is the first card of the lowest possible straight flush? You would have to have a 5, 4, 3, 2, and Ace to create the lowest possible flush. Now we need to count all of the possible combinations in between.

Rather than using a formula, the easiest way is to count each card that begins the straight flush. For example, our first straight flush started with a King, which would be one possible combination. The next would start with a Queen (and the hand would be Queen, Jack, 10, 9, and 8). If we continue counting like this, then we have nine possible combinations for each suit. Since there are four suits, we need to multiply 9 x 4 to get all possible combinations of straight flushes, which is 36. Now, we divide that by the number of total possible hands in a deck of 52 cards, and we get 36/2,598,960, or approximately a 1 in 72,193 chance.

A flush is a hand of five cards that are all the same suit. They do not have to be consecutive in order. This is an easier probability to find. There are four suits, and of those four you can only have one in your hand. Therefore, we can look at four possible combinations of flushes from four different suits. Each suit has 13 cards, and we need five of those cards from one of the suits. So we will also need to consider all of the possible ways you can choose five cards from 13 cards in one suit. This gives us the probability combination of 13 choose 5. For this, we will need to use the combination formula again.

combination formula 13 choose 5

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