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”?
a) To calculate the probability of a fatal accident over a lifetime, we can use the concept of independent events.
The probability of a fatal accident on a single auto trip is 1 in 4,000,000, which can be written as 1/4,000,000 or 0.00000025.
The probability of not having a fatal accident on a single auto trip is the complement of the probability of having a fatal accident, which is 1 - 0.00000025 = 0.99999975 or approximately 1.
Since the events are assumed to be independent, the probability of not having a fatal accident on all 50,000 trips can be calculated by multiplying the probability of not having a fatal accident on a single trip, 0.99999975, by itself 50,000 times.
P(not having a fatal accident on all 50,000 trips) = (0.99999975)^50,000 ≈ 0.9933
Therefore, the probability of not having a fatal accident on all 50,000 trips is approximately 0.9933.
To calculate the probability of having at least one fatal accident over a lifetime, we can subtract the probability of not having a fatal accident on all 50,000 trips from 1.
P(having at least one fatal accident over a lifetime) ≈ 1 - 0.9933 ≈ 0.0067
Therefore, the probability of a fatal accident over a lifetime is approximately 0.0067 or 1 in 150.
The assumption of independence might be violated if there are factors that can affect the likelihood of a fatal accident on multiple trips. For example, a driver who has been involved in a near-fatal accident might be more cautious and drive defensively on subsequent trips, reducing the likelihood of future accidents. Likewise, a driver who engages in risky behavior on one trip might be more likely to engage in risky behavior on subsequent trips, increasing the likelihood of future accidents. These factors can introduce dependence between the events and affect the accuracy of the calculation.
b) A driver might be tempted not to use a seat belt "just on this trip" because they might perceive the probability of a fatal accident on that specific trip to be very low. However, it is important to remember that the probability of a fatal accident on a single trip, although low, is still present. By not using a seat belt, the driver increases their risk of severe injuries or death in the event of an accident.
Seat belts are designed to restrain passengers in the vehicle and prevent them from being thrown forward or ejected from the car during a collision. They are highly effective in reducing the risk of serious injuries and fatalities. Even if the probability of a fatal accident on a single trip is small, it is still crucial to prioritize safety and take precautionary measures such as wearing a seat belt on every trip.
It is important to make decisions based on statistics and probabilities rather than relying on luck or perceptions of safety. Seat belts have been proven to save lives, and it is always advisable to use them for protection in any vehicle.