What rule did we use? In given Example, we calculated P(B). What rule did we use for this calculation? Explain why this rule applies in this setting.
Example
College students. Here is the distribution of U.S. college students classified by age and full-time or part-time status:
|
Age
|
Status
|
|
Full-time
|
Part-time
|
|
15 to 19
|
0.21
|
0.02
|
|
20 to 24
|
0.32
|
0.07
|
|
25 to 34
|
0.10
|
0.10
|
|
35 and over
|
0.05
|
0.13
|
Let's compute the probability that a full-time student is aged 15 to 19. We know that the probability that a student is full-time and aged 15 to 19 is 0.21 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 a full-time student
B = the student is between 15 and 19 years of age
Our formula is
P(B | A) = P(A and B) / P(A)
We read P(A and B) = 0.21 from the table as we mentioned previously. What about P(A)? 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(A) = 0.21 + 0.32 + 0.10 + 0.05 = 0.68
We are now ready to complete the calculation of the conditional probability:
P(B | A) = P(A and B) / P(A)
= 0.21/0.68
= 0.31
The probability that a student is 15 to 19 years of age given that the student is full-time is 0.31.