1. A political science class has 20 students, each of whom chooses a topic from a list for a term paper. How big a pool of topics is necessary for the probability of at least one duplicate to drop below 50%?
2. Refer to Exercise 24. How large would the audience have to have been so that the probability that someone in the audience had the same birthday as Carson was at least 50%?
Refer to Exercise 24,
Johnny Carson, host of The Tonight Show from 1962-1992, discussed the birthday problem during one of his monologues. To test the hypothesis, he asked the audience whether any of them were born on his birthday, October 23. Carson was surprised that no one in the audience of 100 people shared his birthday. What was wrong with Carson's reasoning? What is the probability that one or more people in the audience were born on October 23?