Probability of Drawing a Full House

Statistics Tutorial

Probability of Drawing a Full House


We are going to randomly select 5 cards from a standard deck of 52 playing cards. We will find the probability of drawing a full house. First let us define a full house. A full house is a combination of a three of a kind and a pair. Ex: Three aces with two kings


Sample Problem

Find the probability of drawing a full house from a 52 card deck.


First we will assume we are drawing cards randomly from a well shuffled deck of cards.

To solve this, we will need to rely on counting methods. We will also need to find the size of our sample space (all possible 5 card hands) and our event of interest (all possible full houses)

Let the event A be all possible full houses and S be our sample space.
Let’s find S first.

S is all possible 5 card hands. So it is simply 52 choose 5. Or 52!/5!*(52-5)! = 52!/5!*47!

We use the combination counting method since we don’t care about the order in which we select the cards. The permutation method does account for order.

Now find A. This is a little more tricky. Now there are 13 different kinds of cards: 2,3,4,5,6,7,8,9,10,J,Q,K,A. And each kind has 4 suits. To find a full house, we will select the three of a kind first. There are 13 kinds of cards to choose from, so we have 13 choose 1 ways of doing so or simply, 13. We choose one of the 13, like 7 for example. Now, there are 4 different 7’s. So we need to count the ways we can select three of the four 7’s. This would be 4 choose 3. Four 7’s and three ways of selecting them. Now let’s find the pair. There are 12 kinds of cards left to choose from since we selected a kind of card for the three of a kind. There are then 12 choose 1 ways of doing this, suppose we choose the 8. Now find the number of ways to have a pair of 8’s. This would be 4 choose 2. Four 8’s and choose two of them. We now have the parts necessary for the event A. By the m,n rule, we multiply the results;

(13 choose 1)*(4 choose 3)*(12 choose 1)*(4 choose 2) = 13*4*12*6

To find the probability we divide this by S

13*4*12*6/(52 choose 5) = .0014 or .14%

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