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Consider a situation where the population has 100 students. 60 are male and 40 are female. Among the males, 25 wear glasses, while among the females, only 10 wear glasses.

The overall probability of wearing glasses is P(G)=35/100=35%.

What is the probability of wearing glasses for Male Students?

There are 60 males and 25 wear glasses, so P(G|M)=25/60. This is a CONDITIONAL probability -- What is the probability of a student wearing glasses under the condition that the student is a male?

By the same reasoning, P(G|F)=10/40=25%

THE MULTIPLICATION RULE provides a formula which allows calculation of these conditional probabilities.

M=set of Male Students : 60

F= set of Female Students: 40

G= set of students with Glasses: 35

Question: Let R be a randomly drawn student from this population. We learn that the student is a Female. What is the probability that the student wears glasses, given that we know she is female?

ANSWER: Using the formula for conditional probability, P(G|F), we need P(Both G and F). There are 10 students in the intersection of G and F. Divide this by probability of F. There are 40 students in the set F. Therefore: P(G|F)=10/40

EXERCISES: Compute P(F|G) and P(M|G)

This formula can be taken as the definition of conditional probability. There is another useful way to look at this formula:

The joint probability of two events -- being male and wearing glasses -- can be computed by first calculating the probability of being male, then calculating the probability of wearing glasses GIVEN that the student is male. Multiplying these two numbers gives the joint probability. We can also do it in the OTHER order -- first calculate the probability of wearing Glasses, than the probability of being male, given that the student wears glasses. Multiplying these two will give the same joint probability