In each situation, explain whether the selection is made with replacement or without replacement.
a. The three digits in the lottery in Example 7.2 (p. 223).
b. The three students selected to answer questions in Case Study 7.1 (p. 220).
c. Five people selected for extra security screening while boarding a particular flight.
Example 7.2
A Simple Lottery A lottery game that is run by many states in the United States is one in which players choose a three-digit number between 000 and 999. A player wins if his or her three-digit number is chosen. If we assume that the physical mechanism that is used to draw the winning number gives each possibility an equal chance, we can determine the probability of winning. There are 1000 possible three-digit numbers (000, 001, 002, . . . , 999), so the probability that the state picks the player’s number is 1/1000. In the long run, a player should win about 1 out of 1000 times. Note that this does not mean that a player will win exactly once in every thousand plays.
Case Study 7.1
Last week, Alicia went to her physician for a routine medical exam. This morning, her physician phoned to tell her that one of her tests came back positive, indicating that she may have a disease that we will simply call D. Thinking there must be some mistake, Alicia inquired about the accuracy of the test. The physician told her that the test is 95% accurate as to whether someone has disease D or not. In other words, when someone has D, the test detects it 95% of the time. When someone does not have D, the test is correctly negative 95% of the time.
Therefore, according to the physician, even though only 1 out of 1000 women of Alicia’s age actually has D, the test is a pretty good indicator that Alicia may have the disease. Alicia doesn’t know it yet, but her physician is wrong to imply that it’s likely that Alicia has the disease. Actually, her chance of having the disease is small, even given the positive test result. Later in this chapter you will discover why this is true.
Alicia had planned to spend the morning studying for her afternoon statistics class. At the beginning of each class, her professor randomly selects three different students to answer questions about the material. There are 50 students in the class, so Alicia reasons that she is not likely to be selected. Rather than studying, she uses the time to search the Web for information about disease D.
At the statistics class that afternoon, the 50 student names are written on slips of paper and put into a bag. One name is drawn (without replacement) for each of the three questions. Alicia twice breathes a sigh of relief. She is not picked to answer either of the first two questions. But probability is not in Alicia’s favor on this day: She gets picked to answer the third question.
