f the probability that P will live 20 years is 1/2 and that B will live 20 years is 1/2, then what is the probability that both will live 20 years?

1. Break Down the Problem

We need to find the probability that both P and B will live for 20 years. Given that:

• The probability that P will live for 20 years, $P(P) = \frac{1}{2}$
• The probability that B will live for 20 years, $P(B) = \frac{1}{2}$

2. Relevant Concepts

Since the life spans of P and B are assumed to be independent events, we can use the multiplication rule of probabilities for independent events: $P(A \text{ and } B) = P(A) \times P(B)$

3. Analysis and Detail

We will apply this rule to find the combined probability: $P(P \text{ and } B) = P(P) \times P(B) = \frac{1}{2} \times \frac{1}{2}$

4. Verify and Summarize

Calculating the product: $P(P \text{ and } B) = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$ Thus, the probability that both P and B will live for 20 years is $\frac{1}{4}$.

Final Answer

The probability that both P and B will live 20 years is $\frac{1}{4}$.