706 CHAPTER 11 Probability The events in Example 5 are not independent, since the probability of the drawing of the second spade was affected by removing the first spade drawn from the deck. Such events are called dependent events . Experiments done with replacement will result in independent events, and those done without replacement will result in dependent events. For example, suppose at a fundraiser, a raffle is held in which 100 tickets are sold. Three tickets are randomly drawn, one at a time, from a bin without replacement, and the winners each receive a prize. Are the events of determining the three winners independent or dependent events? The tickets are drawn without replacement. Thus, once a ticket is drawn from the bin, the ticket cannot be drawn again a second or third time. Therefore, the events are dependent, since each time a ticket is drawn, the probability changes for the subsequent drawing. In the first drawing, the probability that a specific ticket is drawn is . 1 100 If that specific ticket is not drawn, then in the second drawing, the probability that it is drawn is now . 1 99 If that specific ticket is still not drawn, then in the third drawing, the probability that it is drawn is now . 1 98 In general, in any experiment in which two or more items are selected without replacement, the events will be dependent. The multiplication formula may be extended to more than two events, as illustrated in Example 6. Did You Know? The Birthday Problem A mong 24 people chosen at random, what would you guess is the probability that at least 2 of them have the same birthday? It might surprise you to learn that it is greater than . 1 2 There are 365 days on which the first person selected can have a birthday. That person has a 365/365 chance of having a birthday on one of those days. The probability that the second person’s birthday is on any other day is 364/365. The probability that the third person’s birthday is on a day different from the first two is 363/365, and so on. The probability that the 24th person has a birthday on any other day than the first 23 people is 342/365. Thus, the probability, P, that of 24 people, no 2 have the same birthday is (365/365) (364/365) (363/365) × × × (342/365) 0.462. …× = Then the probability of at least 2 people of 24 having the same birthday is P 1− = 1 0.462 0.538, − = or slightly larger than . 1 2 We will explore the birthday problem further in Exercise 107. Dmitrii Shironosov/123RF Applets Experiment Birthdays Example 6 Medical Research A medical research study on a new medicine for rheumatoid arthritis is being conducted with 25 patients. After the study was concluded, it was determined that 19 patients reacted favorably to the medicine, 2 reacted unfavorably, and 4 were unaffected. If 3 of the patients are randomly selected, determine the probability of each of the following. a) All three reacted favorably. b) The first patient reacted favorably, the second patient reacted unfavorably, and the third patient was unaffected. c) No patient reacted favorably. d) At least one patient reacted favorably. Solution Each time a patient is selected, the number of patients remaining decreases by 1. a) The probability that the first patient reacted favorably is . 19 25 If the first patient reacted favorably, of the 24 remaining patients only 18 patients are left who reacted favorably. The probability of selecting a second patient who reacted favorably is . 18 24 If the second patient reacted favorably, only 17 patients are left who reacted favorably. The probability of selecting a third patient who reacted favorably is . 17 23 ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⋅ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⋅ ⎛ ⎝ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ = ⋅ ⋅ = P P P P three patients reacted favorably first patient reacted favorably second patient reacted favorably third patient reacted favorably 19 25 18 24 17 23 969 2300 b) The probability that the first patient reacted favorably is . 19 25 Once a patient is selected, there are only 24 patients remaining. Two of the remaining 24 patients reacted unfavorably. Thus, the probability that the second patient reacted unfavorably is . 2 24 After the second patient is selected, there are 23 remaining patients, of which 4 were unaffected. The probability that the third patient was unaffected is therefore . 4 23 StatCrunch Instructor Resources for Section 11.5 in MyLab Math • Objective-Level Videos 11.5 • StatCrunch: Experiment: Birthdays • PowerPoint Lecture Slides 11.5 • MyLab Exercises and Assignments 11.5
RkJQdWJsaXNoZXIy NjM5ODQ=