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Supposing that it is 8 to 7 against a pe...

Supposing that it is 8 to 7 against a person who is now 30 years of age living till he is 60 and 2 to 1 against a person who is now 40 living till he is 70 , find the probability that at least one of these persons will be alive 30 years hence .

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To solve the problem, we need to find the probability that at least one of the two persons will be alive 30 years from now. We will denote the probabilities of the two persons living and not living as follows: Let: - \( P_1 \) = Probability that the first person (30 years old) lives for 30 more years (until age 60). - \( Q_1 \) = Probability that the first person does not live for 30 more years. - \( P_2 \) = Probability that the second person (40 years old) lives for 30 more years (until age 70). - \( Q_2 \) = Probability that the second person does not live for 30 more years. ### Step 1: Determine \( P_1 \) and \( Q_1 \) The odds against the first person living until age 60 are given as 8 to 7. This means: - The probability of living \( P_1 = \frac{7}{7 + 8} = \frac{7}{15} \) - The probability of not living \( Q_1 = 1 - P_1 = 1 - \frac{7}{15} = \frac{8}{15} \) ### Step 2: Determine \( P_2 \) and \( Q_2 \) The odds against the second person living until age 70 are given as 2 to 1. This means: - The probability of living \( P_2 = \frac{1}{1 + 2} = \frac{1}{3} \) - The probability of not living \( Q_2 = 1 - P_2 = 1 - \frac{1}{3} = \frac{2}{3} \) ### Step 3: Calculate the probability that at least one person is alive The probability that at least one of them is alive is given by the formula: \[ P(\text{at least one alive}) = 1 - P(\text{both not alive}) = 1 - (Q_1 \cdot Q_2) \] ### Step 4: Calculate \( Q_1 \cdot Q_2 \) Now we substitute the values of \( Q_1 \) and \( Q_2 \): \[ Q_1 \cdot Q_2 = \left(\frac{8}{15}\right) \cdot \left(\frac{2}{3}\right) = \frac{16}{45} \] ### Step 5: Calculate \( P(\text{at least one alive}) \) Now we can find the probability that at least one person is alive: \[ P(\text{at least one alive}) = 1 - \frac{16}{45} = \frac{45 - 16}{45} = \frac{29}{45} \] ### Conclusion Thus, the required probability that at least one of these persons will be alive 30 years hence is: \[ \frac{29}{45} \]
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