The probability that Ram will be alive 30 years hence is `7/11` and Shyam will be alive is `7/10` . What is the probability that both Ram and Shyam will be dead 30 years hence ?

Options

A. `1/11`
B. `12/110`
C. `12/121`
D. None of these

Answer

To find the probability that both Ram and Shyam will be dead 30 years hence, we can follow these steps:

### Step 1: Determine the probabilities of Ram and Shyam being alive.
- The probability that Ram will be alive 30 years hence is given as \( P(R) = \frac{7}{11} \).
- The probability that Shyam will be alive 30 years hence is given as \( P(S) = \frac{7}{10} \).

### Step 2: Calculate the probabilities of Ram and Shyam being dead.
- The probability that Ram will be dead is the complement of the probability that he will be alive:
\[ P(R') = 1 - P(R) = 1 - \frac{7}{11} = \frac{11 - 7}{11} = \frac{4}{11} \]
- The probability that Shyam will be dead is the complement of the probability that he will be alive:
\[ P(S') = 1 - P(S) = 1 - \frac{7}{10} = \frac{10 - 7}{10} = \frac{3}{10} \]

### Step 3: Calculate the probability that both Ram and Shyam will be dead.
- Since the events of Ram and Shyam being dead are independent, we can multiply their probabilities:
\[ P(R' \cap S') = P(R') \times P(S') = \frac{4}{11} \times \frac{3}{10} \]

### Step 4: Perform the multiplication.
- Now, we multiply the fractions:
\[ P(R' \cap S') = \frac{4 \times 3}{11 \times 10} = \frac{12}{110} \]

### Step 5: Simplify the fraction.
- We can simplify \( \frac{12}{110} \):
\[ \frac{12}{110} = \frac{6}{55} \]

### Final Answer:
The probability that both Ram and Shyam will be dead 30 years hence is \( \frac{6}{55} \).

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