# Conditional Probability Examples

## What is Conditional Probability?

When events A and B are dependent, the probability that both A and B occurs is:

P(A and B) = P(A) * P(B|A)

P(B|A) means “the probability that event B occurs, given that event A has occurred.” This is known as a conditional probability because the probability that event B occurs depends on the condition of event A.

The formula to find conditional probability is:

P(B|A) = P(A and B) / P(A)

Events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B)

Example 1: Researchers surveyed 200 students on which movie genre they prefer. The results of the survey are shown in the two-way table below: What is the probability that a randomly chosen student is male?

P(male) = total males / total students = 95 /200 = 0.475

What is the probability that a randomly chosen student prefers comedy?

P(prefers comedy) = total students who prefer comedy / total students = 90 / 200 = 9/20

What is the probability that a randomly chosen student is male, given the student prefers comedy?

P(student is male, given they prefer comedy) = P(male and prefers comedy) / P(prefer comedy)

P(student is male, given they prefer comedy) = (50/200) / (90/200)

P(student is male, given they prefer comedy) = 5/9

What is the probability that a randomly chosen student prefers comedy, given the student is male?

P(prefers comedy, given student is male) = P(prefers comedy and is male) / P(male)

P(prefers comedy, given student is male)= (50/200) / (95/200)

P(prefers comedy, given student is male) = 50/95 = 10/19

Are the events “is male” and “prefers comedy” independent?

Recall that events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B)

In this case, P(prefers comedy, given student is male) = 10/19 and P(prefers comedy) = 9/20. Since these two probabilities are not equal, these two events are not independent.

Example 2: Researchers surveyed residents from two different cities about their annual incomes. The results of the survey are shown in the two-way table below: What is the probability that a randomly selected resident is from city A?

P(resident from city A) = total residents from city A / total residents in survey = 260/500 = 13/25

What is the probability that a randomly selected resident earns more than \$50,000 annually?

P(more than \$50k) = total residents who earn more than \$50k / total residents = 140/500 = 7/25

What is the probability that a randomly selected resident is from city A, given the resident earns more than \$50,000 annually?

P(from city A, given income > \$50k) = P(from city A and income > \$50k) / P(income > \$50k)

P(from city A, given income > \$50k) = (60/500) / (140/500) = 60/140 = 3/7

What is the probability that a randomly selected resident earns more than \$50,000 annually, given the resident is from city A?

P(income > \$50k, given from city A) = P(income > \$50k and from city A) / P(from city A)

P(income > \$50k, given from city A) = (60/500) / (260/500) = 3/13

Are the events “is from city A” and “earns more than \$50,000 annually” independent?

Recall that events A and B are independent if P(A|B) = P(A) and P(B|A) = P(B)

In this case, P(income > \$50k, given from city A) = 3/13 and P(income > \$50k) = 7/25. Since these two probabilities are not equal, these two events are not independent.