Suppose that the probability is 1 in 3,900,000 that a single auto trip in the United States will result in the death of the driver. Over a lifetime, an average U.S. driver takes 50,000 trips. Assume that events are independent: (a) What is the probability of such a driver surviving 50,000 such trips? (b) What is the probability of having a fatal accident over the span of 50,000 trips? (c) Why might the assumption of independence be violated? (d) Why might a driver be tempted not to use a seat belt “just on this trip”?
(a) To find the probability of a driver surviving 50,000 trips, we can use the complement rule.
The probability of survival on a single trip is 1 - (1/3,900,000) = 3,899,999/3,900,000.
Since each trip is considered independent, the probability of surviving all 50,000 trips is calculated by multiplying the probabilities of survival on each trip:
P(surviving 50,000 trips) = (3,899,999/3,900,000)^50,000
(b) The probability of having a fatal accident over the span of 50,000 trips can be calculated using the complement of the probability of survival:
P(fatal accident in 50,000 trips) = 1 - P(surviving 50,000 trips)
(c) The assumption of independence may be violated in certain cases. For example, if a driver is involved in a near-miss accident or a minor accident, it may alter the probability of future accidents. Additionally, external factors like weather conditions, road quality, and other drivers' behavior could also impact the assumption of independence.
(d) A driver might be tempted not to use a seat belt "just on this trip" due to a variety of reasons. These could include overconfidence or a perception that the probability of a fatal accident is so low that taking precautionary measures, such as wearing a seat belt, is unnecessary. However, it's important to remember that the probability of an accident is still non-zero, and the consequences of not wearing a seat belt can be severe. By consistently using a seat belt, a driver ensures their safety in all trips, even in rare and unexpected situations.