The probability that Krishna will be alive 10 years hence is 7/15 and that Hari will be alive is 7/10. What is t he probability that both Krishna and Hari will be dead 10 years hence?

Explanation

To find the probability that both Krishna and Hari will be dead 10 years hence, we can follow these steps:

### Step 1: Determine the probabilities of survival
Let:
- \( P(K) \) = Probability that Krishna will be alive in 10 years = \( \frac{7}{15} \)
- \( P(H) \) = Probability that Hari will be alive in 10 years = \( \frac{7}{10} \)

### Step 2: Calculate the probabilities of being dead
To find the probabilities that they will be dead, we need to calculate the complements of the survival probabilities:
- \( P(K') \) = Probability that Krishna will be dead in 10 years = \( 1 - P(K) = 1 - \frac{7}{15} \)

\[ P(K') = 1 - \frac{7}{15} = \frac{15 - 7}{15} = \frac{8}{15} \]

- \( P(H') \) = Probability that Hari will be dead in 10 years = \( 1 - P(H) = 1 - \frac{7}{10} \)

\[ P(H') = 1 - \frac{7}{10} = \frac{10 - 7}{10} = \frac{3}{10} \]

### Step 3: Calculate the probability that both are dead
Since the events of Krishna being dead and Hari being dead are independent, we can multiply their probabilities:

\[ P(K' \cap H') = P(K') \times P(H') = \frac{8}{15} \times \frac{3}{10} \]

### Step 4: Perform the multiplication
Now, we multiply the two fractions:

\[ P(K' \cap H') = \frac{8 \times 3}{15 \times 10} = \frac{24}{150} \]

### Step 5: Simplify the fraction
To simplify \( \frac{24}{150} \):

\[ \frac{24 \div 6}{150 \div 6} = \frac{4}{25} \]

### Final Answer
Thus, the probability that both Krishna and Hari will be dead in 10 years is:

\[ \frac{4}{25} \]

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