Mortality tables

Mortality tables enable actuaries to obtain the probability that a person at any age will live a specified number of years. Insurance companies and others use such probabilities to determine life-insurance premiums, retirement pensions, and annuity payments. According to tables provided by the National Center for Health Statistics in Vital Statistics of the United States, a person of age 20 years has about an 80% chance of being alive at age 65 years. Suppose three people of age 20 years are selected at random.

a. Formulate the process of observing which people are alive at age 65 as a sequence of three Bernoulli trials.
b. Obtain the possible outcomes of the three Bernoulli trials.
c. Determine the probability of each outcome in part (b).
d. Find the probability that exactly two of the three people will be alive at age 65.
e. Obtain the probability distribution of the number of people of the three who are alive at age 65.

Correct answer:

d =  0.384

Step-by-step explanation:

$$\begin{gathered} p_1=80\mathrm{\%}={\displaystyle \frac{80}{100}}=0.8 \\ d=\left(\genfrac{}{}{0px}{}{3}{2}\right)\cdot p_1^2\cdot (1-p_1{)}^{3-2}=\left(\genfrac{}{}{0px}{}{3}{2}\right)\cdot 0.8^2\cdot (1-0.8{)}^{3-2}=0.384 \end{gathered}$$

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