In a test for ESP, a picture is randomly selected from four possible choices and hidden away. Participants are asked to describe the hidden picture, which is unknown to anyone in the environment. They are later shown the four possible choices and asked to identify the one they thought was the hidden target picture. Because the correct answer had been randomly selected from the four choices before the experiment began, the probability of guessing correctly by chance alone is 1/4 or .25. Each time the experiment is repeated, four new pictures are used.
a. Researchers are screening people for potentially gifted ESP participants for a later experiment. They will accept only people who get three correct answers in three repetitions of the screening experiment. What is the probability that someone who is just guessing will be accepted for the later experiment?
b. Refer to part (a). Suppose that an individual has some ESP ability and can correctly guess with probability .40 each time. What is the probability that such a person will be accepted for the later experiment?
c. Refer to parts (a) and (b). Suppose that half of the people in the population being tested are just guessing and have no ESP ability, while the other half can actually guess correctly with probability .40 each time. Given that someone is accepted for the upcoming experiment, what is the probability that the person actually has ESP ability?