Answer.....

A black box f has either a four sided die marked 1 through 4, or four coins. If f is four coins, then each time f is activated, it ips the coins and reports how many heads there are (the probability distribution in this case is that the result is 0, 1, 2, 3, 4 with probability 1/16, 1/4, 3/8, 1/4, 1/16). If a distinguishing algorithm were to invoke the black box once and output "coins" if the result is O and "die" otherwise, what advantage would it achieve? Write your answer as a reduced fraction without any spaces (for example: 7/39). But, a more clever distinguishing algorithm could get a higher advantage. What is the maximum advantage achievable when activating the black box only one time? Write your answer as a reduced fraction without any spaces (for example: 7/39)