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Answer: 42%
## Explanation This is a conditional probability problem. We need to find the probability that a beneficiary will receive a lump-sum payment **given that** they have already opted for monthly disbursements. ### Given Data: - Total beneficiaries: 100 - Monthly disbursements (A): 57 - Lump-sum payment (B): 43 - Both monthly and lump-sum (A ∩ B): 24 ### Applying Conditional Probability Formula: \[ P(B \mid A) = \frac{P(A \cap B)}{P(A)} \] Where: - \[ P(A \cap B) = \frac{24}{100} = 0.24 \] - \[ P(A) = \frac{57}{100} = 0.57 \] ### Calculation: \[ P(B \mid A) = \frac{0.24}{0.57} = 0.4211 \approx 42\% \] ### Why Other Options Are Incorrect: - **A (24%)**: This is simply \[ P(A \cap B) \], the probability of both events occurring, not the conditional probability. - **C (50%)**: This appears to be an average of the two probabilities, not the correct conditional probability calculation. - **D (56%)**: This would be \[ P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.24}{0.43} \approx 56\% \], which is the reverse conditional probability (probability of monthly given lump-sum). Therefore, the correct answer is **42%**.
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A risk manager at a civil service pension scheme is conducting a year-end review on the disbursement preferences of 100 retirement plan beneficiaries and observes the following:
If a retirement plan beneficiary selected at random from this sample has opted for monthly disbursements, what is the probability that the beneficiary will also receive a lump-sum payment?
A
24%
B
42%
C
50%
D
56%
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