In an ESP test, a "participant" tries to draw a hidden "target" photograph that is unknown to anyone in the room. After the drawing attempt, the participant is shown four choices and asked to determine which one had been the real target. The real target is randomly selected from the four choices in advance, so the probability of a correct match by chance is 1/4. The test is repeated ten times, and four new photographs are used each time.
a. Go through the conditions for a binomial experiment, and explain how this situation fits each one of them, assuming that the participant is just guessing each time.
b. Let X = number of correct choices in the ten tests. If the participant is just guessing, is X a binomial random variable? If not, explain why not. If so, specify n and p.
c. If the participant is just guessing, find P(X = 6).
d. Suppose the participant actually has some psychic ability and can get each answer correct with probability .5 instead of .25. Find P(X = 6).
e. Compare the answers in parts (c) and (d). If the participant actually selects six of the ten answers correctly, would you believe that he or she was just guessing or that he or she was using some psychic ability? Explain your answer. (Note that there is no correct answer here; your reasoning is what counts.)