In a test for extrasensory perception, the participant repeatedly guesses the suit of a card randomly sampled with replacement from an ordinary deck of cards. There are four suits, all equally likely. The participant guesses 100 times, and X = number of correct guesses.
a. Explain why X is a binomial random variable, and specify n and p, assuming that the participant is just guessing.
b. Find the mean and standard deviation for X if the participant is just guessing.
c. Suppose that the participant guesses correctly 33 times. Find the approximate probability of guessing this well or better by chance.
d. Suppose that the participant guesses correctly 50 times. Find the approximate probability of guessing this well or better by chance. In that circumstance, would you be convinced that the participant was doing something other than just guessing? Explain.