The probability is 1 in 4,000,000 that a single auto trip in the United States will result in a
fatality. Over a lifetime, an average U.S. driver takes 50,000 trips. (a) What is the probability of
a fatal accident over a lifetime? Explain your reasoning carefully. Hint: Assume independent
events. Why might the assumption of independence be violated? (b) Why might a driver be
tempted not to use a seat belt “just on this trip”?
(a) To find the probability of a fatal accident over a lifetime, we can calculate the probability of not having a fatal accident on each trip and then subtract it from 1.
The probability of not having a fatal accident on a single trip is 1 minus the probability of having a fatal accident. In this case, the probability of not having a fatal accident on a single trip is 1 - (1/4,000,000) = 3,999,999/4,000,000.
Since each trip is assumed to be independent, we can multiply the probabilities together for all 50,000 trips. So the probability of not having a fatal accident over a lifetime of 50,000 trips is (3,999,999/4,000,000)^50,000.
Finally, we subtract this probability from 1 to find the probability of having at least one fatal accident over a lifetime:
Probability of a fatal accident = 1 - (3,999,999/4,000,000)^50,000
(b) A driver might be tempted not to use a seat belt "just on this trip" because they believe that the probability of a fatal accident on a single trip is already low, at 1 in 4,000,000. However, it's important to note that each trip is independent and the use of a seat belt reduces the risk of injury or fatality in the event of an accident. Therefore, even with a low probability, wearing a seat belt should be considered as an additional safety measure.
It's also worth mentioning that the assumption of independence might be violated in certain situations. For example, if a driver is fatigued, under the influence of drugs or alcohol, distracted, or driving in adverse weather conditions, the independence assumption may not hold true. These factors can increase the risk of accidents and the severity of the outcome, making it even more important to take precautions such as using a seat belt.