5.70 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”?

5/400 = 1/80

http://www.members.cox.net/dagershaw/lol/Odds.html

I hope this helps.

To calculate the probability of a fatal accident over a lifetime, we need to use the concept of independent events.

(a) The probability of a fatal accident on a single trip is given as 1 in 4,000,000. Assuming independence of each trip, we can calculate the probability of surviving a single trip as 1 - (1/4,000,000) = 3,999,999/4,000,000.

Since we assume independence, the probability of surviving all 50,000 trips is calculated by multiplying the probability of surviving each trip. The probability of surviving all trips is ((3,999,999/4,000,000) ^ 50,000) ≈ 0.9875.

Therefore, the probability of at least one fatal accident over a lifetime is 1 - 0.9875 ≈ 0.0125, or approximately 1.25%.

The assumption of independence might be violated in this case if there are factors that could affect the likelihood of a fatal accident during multiple trips, such as driving conditions, location, or personal behavior. For example, if a driver is fatigued, under the influence of drugs or alcohol, or prone to reckless driving, these factors could increase the probability of a fatal accident across multiple trips.

(b) A driver might be tempted not to use a seat belt "just on this trip" due to a misconception called "gambler's fallacy." The gambler's fallacy refers to the belief that if a certain outcome has not occurred recently, it is more likely to occur soon. This fallacy leads to the false belief that not wearing a seat belt on a particular trip will decrease the chances of a fatal accident because it did not happen previously.

However, it is important to remember that each trip is an independent event, and the probability remains the same for each trip. Not using a seat belt significantly increases the risk of fatality in case of an accident, regardless of the outcome of previous trips. Seat belts are designed to save lives and reduce the risk of serious injury, so it is crucial to always wear one for personal safety.