Example 3.12 from Statistics, 3.2 Independent and Mutually Exclusive Events

Independent and Mutually Exclusive Events: example 3.12

In a particular class, 60 percent of the students are female. Fifty percent of all students in the class have long hair. Forty-five percent of the students are female and have long hair. Of the female students, 75 percent have long hair. Let F be the event that a student is female. Let L be the event that a student has long hair. One student is picked randomly. Are the events of being female and having long hair independent?

The following probabilities are given in this example:

P(F) = 0.60; P(L) = 0.50
P(F AND L) = 0.45
P(L|F) = 0.75

NOTE

The choice you make depends on the information you have. You could use the first or last condition on the list for this example. You do not know P(F|L) yet, so you cannot use the second condition.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution 1
Check whether P(F AND L) = P(F)P(L). We are given that P(F AND L) = 0.45, but P(F)P(L) = (.60)(.50) = .30. The events of being female and having long hair are not independent because P(F AND L) does not equal P(F)P(L).
Solution 2
Check whether P(L|F) equals P(L). We are given that P(L|F) = .75, but P(L) = .50; they are not equal. The events of being female and having long hair are not independent.
Interpretation of results
The events of being female and having long hair are not independent; knowing that a student is female changes the probability that a student has long hair.

0 of 3 steps open

How did it go?

NOTE

Solution 1

Solution 2

Interpretation of results

NOTE

Solution 1

Solution 2

Interpretation of results