Exercise 17.7 - Hot Hands, Cold Statistics. Suppose that during any given game, basketball players are in one of the following two states: Hot (make 75% of their shots, inde- pendent of past success) or Normal (make 50% of their shots, independent of past success). a What is the probability that NBA player Paul will make three baskets in a row in the case that he is Hot? What if he is Normal? b If Paul is equally likely to be in either state before the game, what is the probability that he is Hot after he makes his first three baskets in a row? Your friend believes in the law of small numbers. He thinks (erroneously) that if Paul is Hot, he always makes three out of every four shots he takes. Similarly, when Paul is Normal, he makes exactly two out of every four shots. c If Paul is Hot, with what probability does your friend believe that Paul will make his first basket? After making his first basket, what does your friend believe about the probability that Paul will make his next basket? Explain why these probabilities are not the same. 414 Beliefs d If your friend believes that each state is equally likely before the game, what would he believe about the chance that Paul is Hot after he makes his first three baskets in a row? Explain intuitively why your friend's belief differs from the correct probability that you calculated in part (b).