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Events A and B are independent and have equal probabilities

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Events A and B are independent and have equal probabilities

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Events A and B are independent and have equal probabilities of occurring. What is the probability that event A occurs?

(1) The probability that at least one of events A and B occurs is 0.84.
(2) The probability that event B occurs and event A does not is 0.24.

OA A

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BTGmoderatorDC wrote:
Events A and B are independent and have equal probabilities of occurring. What is the probability that event A occurs?

(1) The probability that at least one of events A and B occurs is 0.84.
(2) The probability that event B occurs and event A does not is 0.24.
Statement 1:
P(at least one event occurs) = 1 - P(neither event occurs)
Thus:
P(at least A or B) = 1 - P(neither A nor B)
Since P(at least A or B) = 0.84, we get:
0.84 = 1 - P(neither A nor B)
P(neither A nor B) = 0.16
P(not A) * P(not B) = 0.16.

Since it is given that P(A) = P(B), P(not A) = P(not B).
Substituting P(not A) = P(not B) into P(not A) * P(not B) = 0.16, we get:
[P(not A)]² = 0.16
P(not A) = 0.4
P(A) = 1 - 0.4 = 0.6
SUFFICIENT.

Statement 2:
Try to recycle the result yielded by Statement 1:
P(A) = P(B) = 0.6
Case 1: P(B) = P(A) = 0.6 and P(not A) = 0.4, with the result that P(B but not A) = (0.6)(0.4) = 0.24
The values in blue can be reversed, as follows:
Case 2: P(B) = P(A) = 0.4 and P(not A) = 0.6, with the result that P(B but not A) = (0.4)(0.6) = 0.24
Since P(A) can be different values, INSUFFICIENT.

The correct answer is A.

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