One classical experiment on ESP (extrasensory perception) tests for the ability of an individual to show telepathy—to read the mind of another individual. This test uses five cards with different designs, all known to both participants. In a trial, the “sender” sees a randomly chosen card and concentrates on its design. The “receiver” attempts to guess the identity of the card. Each of the five cards is equally likely to be chosen, and only one card is the correct answer at any point.

a. Out of 10 trials, a receiver got four cards correct. What is her success rate? What is her expected rate of success, assuming she is only guessing?

b. Is her higher actual success rate reliable evidence that the receiver has telepathic abilities? Carry out the appropriate hypothesis test.

c. Assume another (extremely hypothetical) individual tried to guess the ESP cards 1000 times and was correct 350 of those times. This is very significantly different from the chance rate, yet the proportion of her successes is lower than the individual in part (a). Explain this apparent contradiction.