EXAMPLE 4.45

College students. Here is the distribution of U.S. college students classified by age and full-time or part-time status:

Let’s compute the probability that a student is aged 20 to 24, given that the student is full-time. We know that the probability that a student is part-time and aged 20 to 24 is 0.32 from the table of probabilities. But what we want here is a conditional probability, given that a student is full-time. Rather than asking about age among all students, we restrict our attention to the subpopulation of students who are full-time. Let

A = the student is between 20 and 24 years of age

B = the student is a full-time student

Our formula is

We read P(A and B) = 0.32 from the table as we mentioned previously. What about P(B)? This is the probability that a student is full-time. Notice that there are four groups of students in our table that fit this description. To find the probability needed, we add the entries:

P(B) = 0.21 + 0.32 + 0.10 + 0.05 = 0.68

We are now ready to complete the calculation of the conditional probability:

= 0.47

The probability that a student is 20 to 24 years of age, given that the student is full-time, is 0.47.