### 8.4 Expecting the Unexpected

• Math Help

Here are some additional steps for the solution to Example 4.

1. The probability that the 65-year-old man lives to age 80 is 59% = 0.59. So, the probability that he does not live to age 80 is 1 - 0.59 = 0.41. The probability that the 65-year-old female lives to age 80 is 70% = 0.70. So, the probability that she does not live to age 80 is 1 - 0.70 = 0.30. The probability that both do not survive to age 80 is

Probability both do not survive to age 80 = (0.41)(0.30) = 0.123.

Because you know the probability that both do not live to age 80, you can now find the probability that at least one does live to age 80 by subtracting from 1.

Probability at least one does live to age 80 = 1 - 0.123 = 0.877

So, there is an 87.7% chance that at least 1 will survive to age 80.

2. The probability that the 65-year-old man lives to age 90 is 19% = 0.19. So, the probability that he does not live to age 90 is 1 - 0.19 = 0.81. The probability that the 65-year-old female lives to age 90 is 30% = 0.30. So, the probability that she does not live to age 90 is 1 - 0.30 = 0.70. The probability that both do not survive to age 90 is

Probability both do not survive to age 90 = (0.81)(0.70) = 0.567.

Because you know the probability that both do not live to age 90, you can now find the probability that at least one does live to age 90 by subtracting from 1.

Probability at least one does live to age 90 = 1 - 0.567 = 0.433

So, there is a 43.3% chance that at least 1 will survive to age 90.

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• Checkpoint Solution
1. The probability that both women do not survive to age 80 is The probability that at least 1 woman survives to age 80 is So, there is a 91% chance that at least 1 woman will survive to age 80.

The probability that both women do not survive to age 90 is The probability that at least 1 woman survives to age 90 is So, there is a 51% chance that at least 1 woman will survive to age 90.

2. The probability that both men do not survive to age 80 is The probability that at least 1 man survives to age 80 is So, there is an 83.19% chance that at least 1 man will survive to age 80.

The probability that both men do not survive to age 90 is The probability that at least 1 man survives to age 90 is So, there is a 34.39% chance that at least 1 man will survive to age 90.

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