[NW] Life Expectancy
The probability that a randomly selected 40-year-old male will live to be 41 years old is 0.99757, according to the National Vital Statistics Report, Vol. 56, No. 9.


c. What is the probability that at least one of five randomly selected 40-year-old males will not live to be 41 years old? Would it be unusual if at least one of five randomly selected 40-year-old males did not live to be 41 years old?

Verified step by step guidance

1. Identify the probability that a 40-year-old male will not live to be 41 years old. Since the probability of living to 41 is given as \$0.99757\(, the probability of not living to 41 is \)1 - 0.99757$.
2. Define the event of interest: "at least one of five males does not live to be 41." This is the complement of the event "all five males live to be 41."
3. Calculate the probability that all five males live to be 41 by raising the individual survival probability to the power of 5: \((0.99757)^5\).
4. Use the complement rule to find the probability that at least one male does not live to be 41: \$1 - (0.99757)^5$.
5. To determine if this event is unusual, compare the calculated probability to a common threshold for unusual events (often 0.05). If the probability is less than this threshold, the event is considered unusual; otherwise, it is not.

Key Concepts

Complement Rule

The complement rule states that the probability of an event not occurring is one minus the probability that it does occur. In this problem, the probability that a 40-year-old male does not live to 41 is the complement of living to 41, calculated as 1 minus 0.99757.

Independent Events

Independent events are those whose outcomes do not affect each other. Here, the survival of each of the five males is assumed independent, allowing us to multiply probabilities when calculating the chance that all survive or at least one does not.

Probability of 'At Least One' Event

The probability that at least one event occurs among several trials is found by subtracting the probability that none occur from 1. For five males, the chance that at least one does not survive is 1 minus the probability that all five survive.