The odds are 9:5 against a person who is 50 years living till he is 70 and 8:6 against a person who is 60 living till he is 80. Find the probability that at least one of them will be alive after 20 years:a)11/14b)22/49c)31/49d)35/49Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Given information:
- The odds against a 50-year-old person living till 70 are 9:5.
- The odds against a 60-year-old person living till 80 are 8:6.

To find: The probability that at least one of them will be alive after 20 years.

Solution:
Let's first convert the odds into probabilities:
- The probability of a 50-year-old person living till 70 is 5/14 (since odds against are 9:5, probability against is 9/(9+5)=5/14, so probability in favor is 1-5/14=9/14).
- The probability of a 60-year-old person living till 80 is 6/14 (since odds against are 8:6, probability against is 8/(8+6)=4/7, so probability in favor is 1-4/7=3/7).

Now, we can use the formula for the probability of at least one of two events occurring:
P(A or B) = P(A) + P(B) - P(A and B)

- Let A be the event of the 50-year-old person living till 70, and B be the event of the 60-year-old person living till 80.
- The probability of both A and B happening is P(A and B) = P(A) x P(B) = (5/14) x (6/14) = 15/98.
- Therefore, the probability of at least one of them happening is:
P(A or B) = P(A) + P(B) - P(A and B) = (5/14) + (6/14) - (15/98) = 31/49.

Therefore, the probability that at least one of them will be alive after 20 years is 31/49, which is option (c).

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