In this lesson, you'll learn how to use the combination formula to find the probability of getting some of the lower hands in poker. You'll also test your knowledge by calculating the expected values of additional lower hands in the lesson quiz.
Probability in Poker
Angel is playing poker with her friends. Each person bets they have the best hand. Angel has a pair of aces. What is the probability that Angel's friends have higher hands? You've probably played cards and wondered the same thing. In this lesson, we'll look at the probabilities of some of the lower hands in poker, like pairs of cards. We will also find the probabilities of slightly higher hands, such as a full house and four-of-a-kind.
We have a formula to help find these probabilities. The combination formula gives us the number of ways you can choose members from a bigger group.
The total number of values in the group is n, and x is the number of members we want to choose. We often just call this: n choose x. Each exclamation point represents a factorial, which is a number multiplied by all of the integers below it to 1. For example, 3! means 3*2*1.
Among its many uses, the combination formula allows you to calculate the number of all possible hands in poker. We know that there are 52 cards in a deck (not counting jokers), and that you're dealt five cards. Putting those numbers into the combination formula gives us 52!/(52-5)!5!
Let's see how this formula works. There are 52 possible choices for that first card being dealt, 51 choices for the second card, and so on until we get to the fifth card. That gives us 52*51*50*49*48 possible choices.
Using factorials, we can write this as 52!/47!
However, this number over counts the number of possible poker hands. That's because the order in which your five cards were dealt doesn't change your hand. For example, Angel has a pair of aces and three other cards in her hand no matter what order they were dealt in. Rearranging the order of her cards doesn't change the hand she's been dealt.
How many ways can five cards in a hand be rearranged? There are five ways for the first card, four ways for the second card, and so on, which gives us 5*4*3*2*1, or 5!
We need to divide that first value (52/47!) by 5! so we don't count the same hands as different hands. This gives us 52!/47!5!, the same value you'll get from plugging the numbers into the combination formula.
Now let's discuss the probability of each type of poker hand.
Probability of a Full House
A full house is a poker hand that includes a three-of-a-kind, or three cards that are all the same value, such as three jacks or three 4s. A full house also includes a pair of cards, such as two aces or two 7s.
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:
Now, let's look at each individual card in this hand and the probability of getting a three-of-a-kind before we discuss the probability of a pair of cards next.
Remember, there are 13 different values of cards in one suit. To get a three of a kind, you must get the same value for each card. Therefore, the number of ways of getting a certain value for each card is 13 choose 1.
We also need to consider that for a three-of-a-kind we will have three out of the four suits. For example, we might have a 7 of spades, a 7 of clubs, and a 7 of diamonds. Therefore there will be 4 choose 3 combinations to consider for the different suits.
By multiplying the combinations for 13 choose 1 and 4 choose 3, we can get our total number of combinations for a three-of-a-kind for this hand. Next, we need to have a pair of cards as our two remaining cards in the full house.
We must consider that we already have three cards in our hand and don't want to get the same value for any of the remaining cards, because that could give us four of a kind. Therefore, the number of possible values for the other two cards would be 12 choose 1, because we have 12 remaining values to choose from. Next, we need to consider the combinations of suits in the deck once more. This time we have 4 choose 2, because there will be only two cards in this pair.
Using the numbers we got from all of the different combinations, 13, 4, 12, and 6, we can multiply these combinations together to get a total of 3,744 possible combinations for a full house. Dividing by the total number of hands in the deck, you have approximately a 1 in 694 chance of getting a full house.
That's not a bad probability! Next, let's look at the probability of getting a four-of-a-kind.
Probability of Four of a Kind
A four-of-a-kind is a poker hand where a value from all four suits is present. For example, Angel could have four aces or four threes in her hand. Four-of-a-kind is rare and considered a relatively high hand. However, we discuss four-of-a-kind in this lesson because you will see similarities between finding the probability of a four-of-a-kind, three-of-a-kind, and pairs.
To start, we know we will have a value from each of the suits. Remember there are 13 values, or different types of cards in each suit. We will start our combination calculations with 13 choose 1 as we will need one of the values out of 13 for our hand. Next, because we will need all four of each suit, we will need to multiply by 4 choose 4, because there are four suits and we will have one card for each suit.
We also need to consider the 5th card in your hand to account for the total number of possible hands. So, our last part of the calculation will be 48 choose 1, since after the four-of-a-kind we will have 48 cards left in the deck.
The probability of one of Angel's friends getting a four-of-a-kind is approximately 1 in 4,165. However, it is more likely that they'll have two pairs. Let's take a look.
Probability of Pairs
A pair is the most likely poker hand. And because you have five cards, it's possible to get two pairs of cards. So, what is the likelihood that someone has two pair in this card game?
Well, you should know that we will need a 13 choose 2, since we are choosing two different values for the pairs in the hand. Then, we need to consider the combinations of suits 4 choose 2, twice.
Lastly, we need to consider that last card, 11 choose 1 (it's 11 because we've already considered 2 values out of the deck). And for that last card, we need one out of four, or 4 choose 1 for the suits. We have to use this method instead of 48 choose 1 because there are still values left in the deck that we already have in the pairs. For example, if we had a pair of aces and a pair of jacks, we need to consider that there are still two aces and two jacks in the deck; we can't get one of those for the last card because then we would have a full house. So, we choose 11 choose 1 and 4 choose 1 for the last card.
As you can see, Angel is thinking there is a one in 21 chance that someone will have two pair at Angel's table. Angel feels like she has a good chance of winning this hand. How about you? Ultimately, Angel won her poker hand. Her friend either had a high card or a pair, but since Angel had aces, the highest pair you can have, she had the highest hand.
Let's review what we've learned. Many of these probabilities had one thing in common: the combination formula, which is a probability formula that uses factorials to find the number of possible combinations of all the outcomes in the experiment.
In this lesson, we used the combination formula with what we knew about a fair deck of playing cards to find probabilities of different poker hands. However, we've only covered some of the lower hands of poker. Remember, there are a lot of different hands in poker and each has its own way of solving for the probability of the hand.