**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.