Probability and Playing Cards: Playing Poker

Have you ever played poker? How about any card games that involve five cards? 

In this lesson we’ll be exploring the probabilities involving poker. Let’s first start with basic game rules.

Poker Rules

There are many variations to the basic game and for this lesson we’ll be following these rules:

1. We’ll be using a deck of 52 cards. So no joker cards will be used.
2. All cards are unique and none have additional capabilities. For example, sometimes the cards with the number 2 (deuces) are said to be “wild,” meaning that a 2 can become another card. None of that in this simplified game.
3. Five cards will be dealt out.
4. You can discard up to four cards and get new ones.

The rules are important, since we will be using the numerical information that comes from these rule to calculate the probabilities.

When playing poker, or any card game for that matter, someone is usually designate the “dealer.” This is the person who shuffles the deck between games. Shuffling ensures that the cards are randomly arranged from game to game.

The Universe of Shuffled Cards

A shuffled deck is a randomly organized deck. But with 52 cards, how random is the arrangement? Let’s create an arrangement of 52 cards.

With the first card in the arrangement there are 52 possibilities. With the first card chosen, there are 51 possibilities for the second card. Then there are 50 possibilities for the third card. This table summarizes the possibilities.

 

The total number of possible arrangement of cards is the product of all these possibilities:

The total number of possible card shuffles for the deck is 52! (pronounced “52 factorial”). How big is this number? Another way of writing 52! is this way: 

For simplicity, we can write it this way: 

This number is so large, that every time you shuffle a deck of cards, that arrangement of cards is a unique event in the history of the universe!

Watch this video to learn more about the vastness of this number.

The Sample Space of a Poker Hand

The possible arrangement of the 52 cards when the deck is shuffled is enormous. But when playing poker, there are five cards selected out of the 52. How many possible arrangements of five cards are there?

Start with a shuffled deck. 

• Draw the first card out of a possible 52. 
• Draw the second card from a possible 51. 
• Draw the third card from a possible 50. 
• Draw the fourth card from a possible 49. 
• Draw the fifth card from a possible 48.

The total number of possible arrangements is this:

But this calculation assumes that these arrangement of cards are different:

But in poker, these card arrangements are the same hand of cards, so the total number of possible arrangements is overcounting for these variations. We need to adjust the original calculation to divide out these extra arrangements. 

With five cards there are 5! possible arrangements. Revise the calculation this way:

As you can see there are over 2.5 million possible arrangements of five cards from a deck of 52.

Going back to the original calculations, we can rewrite them this way:

The first calculation is called a permutation. In a permutation the order of the five cards matters. As we saw, in poker the arrangement of the cards doesn’t matter. So, the second calculation, called a combination, is the actual number of possible arrangements of five cards in a round of poker. In a combination the order of the cards doesn’t matter.

So, in calculating probabilities involving five-card poker, we are dealing with a sample space of 2,598,960 possible combinations.

Permutations and Combinations

In the previous section you saw calculations for card arrangements that resulted in permutation and combination calculations. In general, these are the formulas to use to calculate permutations and combinations.
 

You’ll sometimes see permutations and combinations written using different notation:

Calculating Permutations and Combinations

A scientific calculator usually has permutation and combination calculation capabilities. You can also use the scientific calculator on Desmos.com.

As you work through the next lessons on probabilities with poker, you will need to use a calculator that can compute permutations and combinations.