Probability that a man who is 40 year old, living till 75 years is `5/16`, and another man who is 35 years old living till 70 years is `3/7` then what is the probability that at least one of them will be alive till 35 years hence?

Options

A. `11/28`
B. `19/28`
C. `17/28`
D. None of these

Answer

To solve the problem step by step, we need to find the probability that at least one of the two men will be alive after 35 years.

### Step 1: Identify the probabilities of living for each man.

- The probability that the 40-year-old man lives until 75 years is given as \( P(A) = \frac{5}{16} \).
- The probability that the 35-year-old man lives until 70 years is given as \( P(B) = \frac{3}{7} \).

### Step 2: Calculate the probabilities of dying for each man.

- The probability that the 40-year-old man dies in the next 35 years is:
\[ Q(A) = 1 - P(A) = 1 - \frac{5}{16} = \frac{11}{16} \]

- The probability that the 35-year-old man dies in the next 35 years is:
\[ Q(B) = 1 - P(B) = 1 - \frac{3}{7} = \frac{4}{7} \]

### Step 3: Calculate the probability that both men die.

The probability that both men die is the product of their individual probabilities of dying:
\[ P(\text{both die}) = Q(A) \times Q(B) = \frac{11}{16} \times \frac{4}{7} \]

Calculating this gives:
\[ P(\text{both die}) = \frac{11 \times 4}{16 \times 7} = \frac{44}{112} = \frac{11}{28} \]

### Step 4: Calculate the probability that at least one of them is alive.

The probability that at least one of them is alive is given by:
\[ P(\text{at least one alive}) = 1 - P(\text{both die}) = 1 - \frac{11}{28} \]

Calculating this gives:
\[ P(\text{at least one alive}) = \frac{28 - 11}{28} = \frac{17}{28} \]

### Final Answer
Thus, the probability that at least one of them will be alive after 35 years is:
\[ \frac{17}{28} \]

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