## If a man tests negative, what is the probability that he actually has the disease?

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.)

### What is the probability that the parcel was shipped express and arrived the next day?

1.  In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective,….

### Given that the test is positive, what is the probability that the person has the disease?

1.  Refer to Exercise 28. Assume that both inspectors inspect every item and that if an item has no flaw, then neither inspector will detect a flaw. a.  Assume that….

### Find the probability mass function of X

1.  Determine whether each of the following random variables is discrete or continuous. a.  The number of heads in 100 tosses of a coin. b.  The length of a rod….