Probability and Playing Cards: Pocket Aces

What is the probability that two cards drawn at random are aces? 

This phenomenon, known as getting pocket aces, is rare. In this lesson we’ll investigate this probability.

Dependent Events

Recall that the probability of choosing a specific card from the deck is this:

Another way of looking at this is that the odds of getting an ace from the deck is 1:13.

Once you have an ace chosen, what is the state of the deck?

With one card on the table, the deck has 51 cards. With one ace on the table, there are only three aces left.

The probability of getting a second ace is dependent on the outcome of the first ace. The probability of getting a second ace becomes this:

So the odds of getting a second ace is 1:17.

What about the combined probability of getting two consecutive aces?

Compound Probability

Getting pocket aces is an example of a compound probability. And as you saw, this is also an example of a dependent event, getting the second act. Here’s how to calculate a compound probability.

Using the probabilities already calculated, here is the probability of getting pocket aces:

This means that the odds of getting pocket aces is 1:221. This means that if you were to shuffle a deck, draw the first two cards, it would take 221 tries to get pocket aces. 

Let’s test this out. Here is a simulation for drawing two cards at a time in a compound event, with dependency on the second draw.

Here’s how the simulation works:

• Clicking on New Cards draws two new cards simulating drawing one card from the deck of 52 and drawing another card from the remaining 51 cards.
• The Counter keeps track of the number of cards you have drawn.
• The Reset button starts the process again.

Activity

Run a simulation of drawing two cards and repeating until you get pocket aces.

Follow these directions: 

• Click on New Cards.  
• Look at  the two cards drawn. If it isn’t two aces, press New Cards.
• Stop when you draw two aces.
• When you do, make a note of the number of draws it took to get pocket aces in a table.
• Do five trails and fill out the table.
• When you are done, find the average of the number of draws. 

 

Note: You’ll be tempted to click quickly through the card draws. Careful because you may skip over pocket aces. Pause enough between clicks of New Cards to make sure you have analyzed the two cards drawn.

Analysis

What was the average number of draws before you got pocket aces? How does compare to the theoretical probability?