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The probability of A and B is the chance that both events A and B will occur at the same time. For example, the probability of rolling a pair of dice and getting a 6 and a 4 is 1/36, since there is only one combination of dice rolls that will produce this result (6 and 4). Similarly, the probability of flipping a coin and getting heads and tails is also 1/36, since there is only one combination of coin flips that will produce this result (heads and tails).
Given two events, A and B, to “find the probability of A and B” means to find the probability that event A and event B both occur.
We typically write this probability in one of two ways:
- P(A and B) – Written form
- P(A∩B) – Notation form
The way we calculate this probability depends on whether or not events A and B are independent or dependent.
If A and B are independent, then the formula we use to calculate P(A∩B) is simply:
Independent Events: P(A∩B) = P(A) * P(B)
If A and B are dependent, then the formula we use to calculate P(A∩B) is:
Dependent Events: P(A∩B) = P(A) * P(B|A)
Note that P(B|A) is the conditional probability of event B occurring, given event A occurs.
The following examples show how to use these formulas in practice.
Examples of P(A∩B) for Independent Events
The following examples show how to calculate P(A∩B) when A and B are independent events.
Example 1: The probability that your favorite baseball team wins the World Series is 1/30 and the probability that your favorite football team wins the Super Bowl is 1/32. What is the probability that both of your favorite teams win their respective championships?
Solution: In this example, the probability of each event occurring is independent of the other. Thus, the probability that they both occur is calculated as:
P(A∩B) = (1/30) * (1/32) = 1/960 = .00104.
Example 2: You roll a dice and flip a coin at the same time. What is the probability that the dice lands on 4 and the coin lands on tails?
Solution: In this example, the probability of each event occurring is independent of the other. Thus, the probability that they both occur is calculated as:
Examples of P(A∩B) for Dependent Events
The following examples show how to calculate P(A∩B) when A and B are dependent events.
Example 1: An urn contains 4 red balls and 4 green balls. You randomly choose one ball from the urn. Then, without replacement, you select another ball. What is the probability that you choose a red ball each time?
Solution: In this example, the color of the ball that we choose the first time affects the probability of choosing a red ball the second time. Thus, the two events are dependent.
Let’s define event A as the probability of selecting a red ball the first time. This probability is P(A) = 4/8. Next, we have to find the probability of selecting a red ball again, given that the first ball was red. In this case, there are only 3 red balls left to choose and only 7 total balls in the urn. Thus, P(B|A) is 3/7.
Thus, the probability that we select a red ball each time would be calculated as:
P(A∩B) = P(A) * P(B|A) = (4/8) * (3/7) = 0.214.
Example 2: In a certain classroom there are 15 boys and 12 girls. Suppose we place the names of each student in a bag. We randomly choose one name from the bag. Then, without replacement, we choose another name. What is the probability that both names are boys?
Solution: In this example, the name we choose the first time affects the probability of choosing a boy name during the second draw. Thus, the two events are dependent.
Let’s define event A as the probability of selecting a boy first time. This probability is P(A) = 15/27. Next, we have to find the probability of selecting a boy again, given that the first name was a boy. In this case, there are only 14 boys left to choose and only 26 total names in the bag. Thus, P(B|A) is 14/26.
Thus, the probability that we select a boy name each time would be calculated as:
P(A∩B) = P(A) * P(B|A) = (15/27) * (14/26) = 0.299.