## Probability Using Permutations and Combinations

### Learning Outcomes

• Compute a conditional probability for an event
• Use Baye’s theorem to compute a conditional probability
• Calculate the expected value of an event

We can use permutations and combinations to help us answer more complex probability questions.

### examples

A 4 digit PIN number is selected. What is the probability that there are no repeated digits?

In a certain state’s lottery, 48 balls numbered 1 through 48 are placed in a machine and six of them are drawn at random. If the six numbers drawn match the numbers that a player had chosen, the player wins $1,000,000. In this lottery, the order the numbers are drawn in doesn’t matter. Compute the probability that you win the million-dollar prize if you purchase a single lottery ticket. ### Example In the state lottery from the previous example, if five of the six numbers drawn match the numbers that a player has chosen, the player wins a second prize of$1,000. Compute the probability that you win the second prize if you purchase a single lottery ticket.

The previous examples are worked in the following video.

### examples

Compute the probability of randomly drawing five cards from a deck and getting exactly one Ace.

### Example

Compute the probability of randomly drawing five cards from a deck and getting exactly two Aces.

View the following for further demonstration of these examples.

## Birthday Problem

Let’s take a pause to consider a famous problem in probability theory:

Suppose you have a room full of 30 people. What is the probability that there is at least one shared birthday?

Take a guess at the answer to the above problem. Was your guess fairly low, like around 10%? That seems to be the intuitive answer (30/365, perhaps?). Let’s see if we should listen to our intuition. Let’s start with a simpler problem, however.

### example

Suppose three people are in a room.  What is the probability that there is at least one shared birthday among these three people?

Suppose five people are in a room.  What is the probability that there is at least one shared birthday among these five people?

Suppose 30 people are in a room.  What is the probability that there is at least one shared birthday among these 30 people?

The birthday problem is examined in detail in the following.

If you like to bet, and if you can convince 30 people to reveal their birthdays, you might be able to win some money by betting a friend that there will be at least two people with the same birthday in the room anytime you are in a room of 30 or more people. (Of course, you would need to make sure your friend hasn’t studied probability!) You wouldn’t be guaranteed to win, but you should win more than half the time.

This is one of many results in probability theory that is counterintuitive; that is, it goes against our gut instincts.

### Try It

Suppose 10 people are in a room. What is the probability that there is at least one shared birthday among these 10 people?