Law of Small Numbers
Probability information and questions in pics
replacement from a deck of 4 cards. When curry is Normal, the deck has 2 "hit" cards and 2 "miss" cards. When Curry is Cold, the deck has 1 "hit" card and 3 "miss" cards. Explain how her model corresponds to the Law of Small Numbers. (d) Suppose Steph Curry is Hot. According to the fan, what is the probability that Curry will make his first basket? After Curry makes his first basket, what does the fan believe about the probability that Curry will make his next basket? Explain why it is lower than the fan's belief about the probability that curry will make his first basket. (e) According to the fan, what is the probability that Curry will make 3 baskets in a row? (f) What if he is Normal? Cold? If the fan has no idea what state Curry will be in before the game (that is, each state is equally likely), what would she believe about the likelihood that he is Hot after he makes his first 3 baskets in a row? Explain intuitively why the fan's beliefs differ from the normatively correct probability that you calculated above. (g) Suppose that, in reality, there is no such thing as being "Hot" or "Cold." Steph Curry is, in fact, always Normal. Over many games, with what frequencies will Curry score 0, 1, 2, and 3 baskets in his first 3 attempts? Suppose the fan attends many games and observes these frequencies. Explain why the fan would not believe you if you tried to convince her that there is no such thing as being "Hot" or "Cold."2. Another manifestation of the representativeness heuristic is that people believe in the "Law of Small Numbers." (b) Suppose that basketball players are, during any given game, in one of three states: Hot (they make 75% of their shots), Normal (they make 50% of their shots), or Cold (they make only 25% of their shots). Suppose Steph Curry is Hot. What is the probability that he will make 3 baskets in a row? What if he is Normal? Cold? If you have no idea what state he'll be in before the game (that is, each state is equally likely), what would you believe about the likelihood that he is Hot after he makes his first 3 baskets in a row? (c) One of the fans at this game believes in the Law of Small Numbers. She has the wrong model of how likely Curry is to make a basket. Here's how her model works. The fan imagines that there is a deck of 4 cards. When Paul is Hot, 3 of these cards say "hit" on them, and only 1 says "miss." Every time Curry takes a shot, one of these cards is drawn randomly without replacement from the deck, and the outcome is whatever the card says. Therefore, when Curry is Hot, he always makes 3 out of every 4 shots he takes. (When the deck is used up, the 4 cards are replaced, the deck is shuffled, and the process begins again - but that isn't important for this problem.) Similarly, when Curry is Normal or Cold, the outcome of every shot is determined by the draw of a card without
