Questions

Answers are given on the next page.

1. A player in a game show is given the choice of three doors. Behind one door is the Grand Prize (say, a car); behind the other two doors are booby prizes (say, goats). The player picks a door, and the show host peeks behind the doors and opens one of the remaining doors. There is a booby prize behind the door he opened. The host then offers the player either to stay with the door that was chosen at the beginning, or to switch to the other closed door. What gives the player the better chance of winning: to switch doors or to stay with the original choice? Or are the chances equal? (This is the famous Monty Hall problem.)

2. Imagine you toss a coin successively and wait till the first time the pattern HTT appears. For example, if the sequence of tosses was

H H T H H T H H T T H H T T T H T H

then the pattern HTT would appear after the 10th toss. Let A(HTT) be the average number of tosses until HTT occurs, and let A(HTH) be the average number of tosses until HTH occurs. Which of the following is true?

A( HTH) < ( HTT), A(HTH) = A(HTT), A(HTH) > A(HTT).

3. Imagine a test for a certain disease (say, HIV) that is 99% accurate. And suppose a person picked at random tests positive. What is the probability that the person actually has the disease?