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”?
To answer these questions, we can use the concept of probability. Here's how we can find the answers:
(a) The probability of surviving a single trip is 1 - (probability of death in a single trip). In this case, the probability of death in a single trip is 1/3,900,000. Therefore, the probability of surviving a single trip is 1 - 1/3,900,000.
To find the probability of surviving 50,000 such trips, we multiply the probabilities of surviving each trip together since each trip is assumed to be independent. Therefore, the probability of surviving 50,000 trips is (1 - 1/3,900,000) ^ 50,000.
(b) The probability of having a fatal accident over the span of 50,000 trips is simply the complement of the probability of surviving all 50,000 trips. We can calculate it as 1 - (probability of surviving 50,000 trips).
(c) The assumption of independence might be violated in situations where factors affecting the probability of death on one trip could also affect the probability of death on subsequent trips. For example, if a driver is involved in a near-fatal accident on one trip, their driving behavior or vehicle condition might change for subsequent trips, affecting the independence assumption.
(d) A driver might be tempted not to use a seat belt "just on this trip" due to a misconception known as the "gambler's fallacy." This fallacy assumes that because the odds of surviving 50,000 trips are low, using a seat belt on a particular trip won't make a difference. However, each trip is independent, so the overall probability of survival is improved by using a seat belt consistently on every trip.
By considering probabilities and understanding the assumptions made, we can analyze the likelihood of survival and make informed decisions regarding safety precautions.