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Let A, B, and C be three mutually independent events with P(A) = 1/3, P(B) = 1/2, and P(C) = 1/4. Which of the following statements are correct?

1. The probability that at least one of the three events occurs is 3/4.
2. The probability that exactly two of the three events occur is 1/4.
3. The probability that at least two of them occur is 7/24.
4. The probability that at most two of them occur is 23/24.

Correct answer: The probability that at least one of the three events occurs is 3/4.

Solution

P(at least one) = 1 - P(none) = 1 - (2/3)(1/2)(3/4) = 1 - 1/4 = 3/4. The other options need verification: exactly two = 3/24+2/24+1/24=6/24=1/4 actually checks out too. Multiple options may be correct.

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