Probability and Playing Cards: Playing Poker

In the previous lesson, we identified the sample space for a five-card poker game from a deck of 52 cards.

This is the total number of possible five-card combinations.

This is the sample space for calculating probabilities for specific card combinations.

A Pair

In a game of poker, you get five cards. What is the probability that two of the cards will be the same number or face card? In other words, what is the probability of a pair?

We start by looking at a set of 13 cards from one suit. There are 13 possibilities and using combinations notation we get this calculation.

For any particular card, there are two cards out of the four suits possible.

Now let’s look at the three remaining cards.

There are three cards taken out of the remaining twelve cards, as shown below.

For each of these three remaining cards, they are available in four suits. So we need to account for that, too.

Now we can calculate the total possibilities.

Finally, we can calculate the probability of getting a pair of cards:

There is roughly a 42% chance of getting a pair with a poker hand. We can test this out with a deck of playing cards. Or you can use a simulation.

This is a simulation of a deck of cards. It can be used to simulate a full game of poker, but for this lesson, we will just focus on drawing the first five cards.

1. Click on the “Deal Cards” button. You will then see five cards dealt out.
   
2. You can exchange up to four cards by identifying the ones you want to flip. Click on the small green buttons to see the word “Flip.”
   
3. Then click on the “Flip Cards” button to exchange the cards you discarded.
   

For this simulation, you’ll really just be focused on Step 1.

Here is a mobile version of this simulation: click on this link.

 

Activity

Run a simulation of dealing 50 rounds of poker. Keep track of the number of times you got a pair.

Follow these directions: 

• Click on Deal Cards. 
• Look at  the five cards drawn. 
• If there are two identical cards, make a note in the table.
• If there are not two identical cards, make a note in the table. Note: If you get three of a kind, count them as not a pair.
• Click on Reset to start a new round.
• Do a total of fifty trials.

Analysis

Of the 50 rounds of poker, how many had a pair? How does this compare to the theoretical probability? Compare your results to those of a classmate.