You are given a bucket that contains 1000 coins. 999 of these are fair coins, but one of them is a trick coin that always comes up heads. you select one coin from this bucket at random. let T be the event that you select the trick coin. Suppose you flip the coin k times. Let Hk be the event that the coin comes up heads all k times. If you see this occur, what is the conditional probability that you have the trick coin?

In a test for esp (extrasensory perception), a subject is told that cards only the experimenter can see contain either a star, a circle, a wave, or a square. as the experimenter looks at each of 20 cards in turn, the subject names the shape on the card. a subject who is just guessing has probability 0.25 of guessing correctly on each card. a. the count of correct guesses in 20 cards has a binomial distribution. what are n and p? b. what is the mean number of correct guesses in 20 cards for subjects who are just guessing? c. what is the probability of exactly 5 correct guesses in 20 cards if a subject is just guessing?