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 need to use the concept of independent events.
The probability of surviving a single auto trip is given as 1 - 1/4,000,000, which is approximately 0.99999975. Since each trip is assumed to be independent, the probability of surviving all 50,000 trips over a lifetime can be calculated by raising the probability of a single trip outcome to the power of the number of trips:
Probability of surviving all 50,000 trips = (0.99999975)^50,000
Using a calculator, the final probability comes out to be approximately 0.999938.
However, to find the probability of a fatal accident over a lifetime, we need to subtract the probability of surviving all 50,000 trips from 1:
Probability of a fatal accident over a lifetime = 1 - 0.999938
The final probability is approximately 0.000062, or 0.0062%.
The assumption of independence may be violated in certain situations. For example, if there is a systemic issue with a particular vehicle or road conditions, multiple trips by the same driver may not be truly independent events. Additionally, if a driver experiences fatigue, distraction, or impaired driving during multiple trips, the independence assumption may not hold.
(b) A driver might be tempted not to use a seat belt "just on this trip" due to various reasons, such as:
1. Overconfidence: The driver may believe that they are skilled enough to avoid accidents and therefore feel that using a seat belt is unnecessary for a particular trip.
2. Convenience: Putting on a seat belt requires some effort and may be perceived as inconvenient, especially for short trips or when the driver is in a hurry.
3. Perceived Low Risk: If the driver believes that the probability of an accident or a fatal outcome on a particular trip is extremely low, they may be tempted to skip using the seat belt.
However, it's important to note that even though the probability of a fatal accident on a single trip may be low, accidents can still happen unexpectedly. Using a seat belt consistently is crucial because it provides protection and significantly reduces the risk of severe injuries or fatalities in case of an accident.