The probability that a 50-years-old man will be alive at 60 is `0.83` and the probability that a 45-years-old women will be alive at 55 is `0.87.` Then

Options

A. the probability that both will be alive is `0.7221`
B. at least one of them will alive is `0.9779`
C. at least one of them will alive is `0.8230`
D. the probability that both will be alive is `0.6320`

Explanation

To solve the problem, we need to find two probabilities:

1. The probability that both the 50-year-old man and the 45-year-old woman will be alive at their respective ages.
2. The probability that at least one of them will be alive.

Let's denote:
- Event A: The event that the 50-year-old man will be alive at 60.
- Event B: The event that the 45-year-old woman will be alive at 55.

Given:
- P(A) = 0.83 (Probability that the man will be alive at 60)
- P(B) = 0.87 (Probability that the woman will be alive at 55)

### Step 1: Calculate the probability that both will be alive (A ∩ B)

Since the events are independent, we can use the multiplication rule for independent events:
\[ P(A \cap B) = P(A) \times P(B) \]

Substituting the values:
\[ P(A \cap B) = 0.83 \times 0.87 \]

Calculating:
\[ P(A \cap B) = 0.7221 \]

### Step 2: Calculate the probability that at least one will be alive

To find the probability that at least one of them will be alive, we can use the complement rule:
\[ P(\text{at least one alive}) = 1 - P(\text{none alive}) \]

First, we need to calculate the probability that neither is alive:
- The probability that the man is not alive:
\[ P(A') = 1 - P(A) = 1 - 0.83 = 0.17 \]

- The probability that the woman is not alive:
\[ P(B') = 1 - P(B) = 1 - 0.87 = 0.13 \]

Now, since the events are independent:
\[ P(A' \cap B') = P(A') \times P(B') \]
\[ P(A' \cap B') = 0.17 \times 0.13 \]

Calculating:
\[ P(A' \cap B') = 0.0221 \]

Now, substituting back to find the probability that at least one is alive:
\[ P(\text{at least one alive}) = 1 - P(A' \cap B') \]
\[ P(\text{at least one alive}) = 1 - 0.0221 = 0.9779 \]

### Summary of Results:
1. The probability that both will be alive is approximately **0.7221**.
2. The probability that at least one will be alive is approximately **0.9779**.