Determine whether the events are mutually exclusive.
-A study of consumer smoking habits includes 168 people in the 18-22 age bracket (48 of whom smoke), 131 people in the 23-30 age bracket (36 of whom smoke), and 99 people in the 31-40 age bracket (28 of whom smoke). If one person is randomly selected from this sample, find the probability of getting someone who is age 18-22 or does not smoke.
A)1.141
B)0.714
C)0.302
D)0.839

Question

Quizplus · Accepted Answer

The events "being age 18-22" and "does not smoke" are not mutually exclusive because a person can be in the age range of 18-22 and also be a non-smoker. To find the probability of selecting someone who is either in the age range of 18-22 or does not smoke, we first find the total number of people in the sample and the number of people who either are in the age range of 18-22 or do not smoke. The total number of people is 168 + 131 + 99 = 398. The number of people who do not smoke is 398 - (48 + 36 + 28) = 398 - 112 = 286. Since the 48 people who are 18-22 and smoke are already included in the 18-22 age group, we don't add them again. So, the number of people who are either 18-22 or do not smoke is 168 (18-22 age group) + 286 (non-smokers) = 454. However, since our total sample size is only 398, this indicates we've double-counted individuals who are both in the 18-22 age group and do not smoke. To correct this, we subtract the number of non-smokers in the 18-22 age group from our total. There are 168 - 48 = 120 non-smokers in the 18-22 age group. So, the corrected total is 454 - 120 = 334. The probability is then 334 / 398 = 0.839.

Determine Whether the Events Are Mutually Exclusive