Now that you can represent probability in a tree diagram, you can solve probability problems. Begin with independent events.

For any two independent events A and B , the probability that both event A and event B occur is given by the following formula.

 

 

In Unit 1, you learned that the intersection of two sets A and B is denoted A ∩ B . This notation is can be used to rewrite the above formula as follows.

 

 

Examples 1 and 2 illustrate the use of this formula.

 

 

 

At the beginning of a football game a coin is flipped to see which team will receive the ball. At the end of the game, if the score is tied, a second coin flip determines which team receives the ball in overtime. What is the probability that the captain of a football team wins the coin flip both at the beginning of the game and for overtime?

The two coin flips represent independent events because the first flip does not affect the results of the second flip. Therefore, use the formula for probability of independent events. Because a coin has two sides, each outcome has a probability of .

The probability that the captain of the football team wins both coin flips is 0.25 or 25%.