1. Refer to Example 2.26.
a. If a man tests negative, what is the probability that he actually has the disease?
b. For many medical tests, it is standard procedure to repeat the test when a positive signal is given. If repeated tests are independent, what is the probability that a man will test positive on two successive tests if he has the disease?
c. Assuming repeated tests are independent, what is the probability that a man tests positive on two successive tests if he does not have the disease?
d. If a man tests positive on two successive tests, what is the probability that he has the disease?
2. If A and B are independent events, prove that the following pairs of events are independent: A c and B, A and B c , and A c and B c .
3. A fair die is rolled twice. Let A be the event that the number on the first die is odd, let B be the event that the number on the second die is odd, and let C be the event that the sum of the two rolls is equal to 7.
a. Show that A and B are independent, A and C are independent, and B and C are independent. This property is known as pairwise independence.
b. Show that A, B, and C are not independent. Conclude that it is possible for a set of events to be pairwise independent but not independent. (In this context, independence is sometimes referred to as mutual independence.)
