## 18. Expected Lifespan

An ethnographer found that in a certain primitive tribe he studied, the distribution of the lifespan of the tribe members could be described as follows: $25 \%$ of them only lived up to 40 years, $50 \%$ died at 50 years, and $25 \%$ - at 60 years. He then randomly selected two individuals to study in more detail. What is the expected lifespan of the one of the two randomly selected individuals who will live longer?
18. If the one of the two members of the tribe who lives longer reaches 40 years, then obviously both "subjects" will live only to 40 years; the corresponding probability is

$$
25 \cdot 25 \% = 1 / 4 \cdot 1 / 4 = 1 / 16
$$

If the "long-liver" reaches 50 years, this means that either the first of the two will live only 40 years, and the second will live 50, or they will both live 50 years, or finally, the first will live 50 years, and the second will live 40 years. The corresponding probability, as is easy to see, is

$$
\begin{aligned}
& (25 \cdot 50) \% + (50 \cdot 50) \% + (50 \cdot 25) \% = \\
& = 1 / 4 \cdot 1 / 2 + 1 / 2 \cdot 1 / 2 + 1 / 2 \cdot 1 / 4 = 1 / 2
\end{aligned}
$$

Finally, if the one of the two chosen members of the tribe who will live longer dies at 60 years, then neither of the events whose probabilities we have just calculated has occurred. The corresponding probability is

$$
1 - 1 / 16 - 1 / 2 = 7 / 16 \text{. }
$$

The expected average life expectancy is

$$
(1 / 16) \cdot 40 + (1 / 2) \cdot 50 + (7 / 16) \cdot 60 = 53 \text{ years } 9 \text{ months. }
$$