A policy requiring all hospital employees to take lie detector tests may reduce losses
due to theft, but some employees regard such tests as a violation of their rights. To gain
some insight into the risks that employees face when taking a lie detector test, suppose
that the probability is 0.06 that a lie detector concludes that a person is lying who, in fact, is telling the truth and suppose that any pair of tests are independent.
What is the probability that a machine will conclude that each of three employees is
lying when all are telling the truth?
For this one I did (0.06)^3
What is the probability that the machine will conclude that none of the employees is
lying when all are telling the truth?
For this one I did (0.94)^3
What is the probability that a machine will conclude that at least one of the three
employees is lying when all are telling the truth?
For this one I did (0.94)^2(0.6)
I am not 100% sure if I am doing this right. Also, I am pretty bad with probability.
Best Answer
For the first one, the probability is $$ \frac{P(\text{detect lying, telling truth})^3}{P(\text{telling truth})^3}=0.06^3 $$ Similarly the second one is $$ \frac{P(\text{detect telling truth, telling truth})^3}{P(\text{telling truth})^3}=0.94^3 $$ So you got the first two right. But for the third one, it should be $$ 1-P(\text{detect no on is lying|all are telling the truth})=1-0.94^3 $$