Statistics and Data Analysis Professor William Greene Stern

Statistics and Data Analysis Professor William Greene Stern School of Business IOMS Department of Economics

Statistics and Data Analysis Part 3 a – Interesting Probability Puzzles

2 Classic Problems and 1 Intriguing One ¢ The birthday problem ¢ The Monty Hall problem ¢ Halftime winner

The Birthday Problem What is the probability that everyone in this room has a different birthday? (50 people)

The Monty Hall Problem Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2? " Is it to your advantage to switch your choice? Answer: It definitely pays to switch. See “Notes for this class” or browse the 1, 000 plus hits you’ll find if you search for this problem on the web.

Throwing in the Towel Consider a sporting event with two halves. Scores accumulate in each half. The winner is the team with the highest total of the scores for the two halves. (Baseball, Hockey, Football, Basketball, Rugby) No ties (so this doesn’t work well for soccer). The two teams are evenly matched, and they play exactly as hard in the second half as in the first. Given that a team is ahead at the halftime, what is the probability that they will win the game? Intuition (incorrectly) says. 5. (If team A wins the first half, it’s as likely that team B will win the second half. ) The correct answer is. 75! The simple intuition is that it is not sufficient for team B to win in the second half. Team B must win by a higher margin in the second half than team A had in the first half. Since they are evenly matched, that probably is only. 25. Formally, there is a 50% chance that team A will win the second half outright. For any first half margin, say M, since they are evenly matched, there is a 50% chance that each team will exceed that margin, so A wins the game in half of the cases that B wins the second half and all of the cases when A wins the second half. See Jeffrey Simonoff, “Probability – the language of randomness, ” pp. 10 -12.