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The probability of A given B is the probability of event A occurring, given that event B has already occurred. To find the probability, you need to calculate the ratio of the number of favorable outcomes to the total number of outcomes. To better understand how to find probability of A given B, consider the example of rolling two dice, where A is the event of rolling a total of 7 and B is the event of rolling an even number. The probability of A given B would be 1/3, since out of the three possible outcomes (4, 6, 8) only one (6 + 1) adds up to 7.
Given two events, A and B, to “find the probability of A given B” means to find the probability that event A occurs, given that event B has already occurred.
We use the following formula to calculate this probability:
P(A|B) = P(A)*P(B|A) / P(B)
where:
- P(A|B): The probability of event A, given event B has occurred.
- P(B|A): The probability of event B, given event A has occurred.
- P(A): The probability of event A.
- P(B): The probability of event B.
The following examples show how to use this formula in practice.
Example 1: Probability of A Given B (Weather)
Suppose the probability of the weather being cloudy is 40%.
Also suppose the probability of rain on a given day is 20%.
Also suppose the probability of clouds on a rainy day is 85%.
If it is cloudy outside on a given day, what is the probability that it will rain that day?
Solution:
- P(cloudy) = 0.40
- P(rain) = 0.20
- P(cloudy | rain) = 0.85
Thus, we can calculate:
- P(rain | cloudy) = P(rain) * P(cloudy | rain) / P(cloudy)
- P(rain | cloudy) = 0.20 * 0.85 / 0.40
- P(rain | cloudy) = 0.425
If it is cloudy outside on a given day, the probability that it will rain that day is 0.425 or 42.5%.
Example 2: Probability of A Given B (Crime)
Also suppose the probability of a police car driving by is 10%.
Also suppose the probability of a crime causing a police car to drive by is 90%.
If a police car drives by, what is the probability that a crime has been committed?
Solution:
- P(crime) = 0.01
- P(police car) = 0.10
- P(police car | crime) = 0.90
Thus, we can calculate:
- P(crime | police car) = P(crime) * P(police car | crime) / P(police car)
- P(crime | police car) = 0.01 * 0.90 / 0.10
- P(crime | police car) = 0.09
If a police car drives by, the probability that a crime has been committed is .09 or 9%.
Example 3: Probability of A Given B (Baseball)
Suppose the probability of a home run being hit in a baseball game is 5%.
Also suppose the probability of a crowd cheering in a stadium when you walk by is 15%.
Also suppose the probability of a crowd cheering when a home run has been hit is 99%.
If you hear a crowd cheering as you walk by the stadium, what is the probability that a home run has been hit?
Solution:
- P(home run) = 0.05
- P(cheer) = 0.15
- P(cheer | home run) = 0.99
Thus, we can calculate:
- P(home run | cheer) = P(home run) * P(cheer | home run) / P(cheer)
- P(home run | cheer) = 0.05 * 0.99 / 0.15
- P(home run | cheer) = 0.33
If you hear a crowd cheering as you walk by the stadium, the probability that a home run has been hit is 0.33 or 33%.
The following tutorials explain how to perform other calculations related to probabilities: