## Conditional Probability

To reduce theft, the ABCD Company screens all its employees with a lie detector test that is known to be correct 90 percent of the time (for both guilty and innocent subjects). The company decides to fire all employees who fail the test. Suppose 5 percent of the employees are guilty of theft.

a. What proportion of the workers are fired?
b. Of the workers fired, what proportion are actually guilty?
c. Of the workers not fired, what proportion are guilty?
d. What do you think of the company's policy?

a.
5% are guilty, 95% are innocent. Workers failing the test will be fired.

Since the test is correct 90% of the time, the proportion of people failing the test can be calculated below:

P(failing) = 5%*90% + 95%*10%
=0.05*0.9+0.95*0.1
=0.14

b.
P(guilty | failing)=5%*90% / (5%*90% + 95%*10%)
=0.05*0.9/0.14
=0.3214

c.
P(guilty | not failing)=5%*10%/(1-0.14)
=0.05*0.1/0.86
=0.05814

d.
The policy is too harsh. Over two thirds of people fired are actually innocent.