goodyear2013 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.
Since A and B have equal probabilities of occurring (Probability p), they have equal probabilities of not occurring.
Probability that neither of the events occurs is 0.16 (the complementary probability of 0.84, or 1 - 0.84).
Therefore, probability of A not occurring and probability of B not occurring is 0.16.
→ (not p)(not p) = 0.16 → (not p)^2 = 0.16 → p = 0.4
If we know probability that event will not occur, then we know probability that it will occur (0.6 = 60%).
Statement (1): Sufficient.
Statement (2): Probability that event A does not occur is simply (1 – p)
p(1 –p) = 0.24 → (p^2 – p + 0.24) = 0, and factoring for 2 numbers that multiply to 0.24 and add to -1 (implicit coefficient in front of p) gives 2 possible values for p: 0.6 & 0.4.
Insufficient.
Hi, the Statement (1) is not very clear to me. Can anyone explain the exact meaning and why it is Sufficient, please.
(To me, it does not sound like the Official GMAT question.)
Let me take the approach used more often:
Assume that P(A) = x.
Given that P(B) = P(A) = x
P(A and B) = P(A) * P(B) = x^2 (Probability that both occur is product of the two since events are independent)
1: The probability that at least one of events A and B occurs is 0.84.
P(A or B) = P(A) + P(B) - P(A and B) = .84
x + x - x^2 = .84
x^2 -2x +.84 = 0
100x^2 - 200x + 84 = 0
This is a quadratic and when you solve it, you get x = 3/5 or 7/5.
The probability cannot be more than 1 so x must be 3/5 = 0.6
Sufficient.
The method used in the explanation is this: P(A or B) = .84
This means that probability that neither occurs = 1 - P(A or B) = .16
P('not A' and 'not B') = P(not A)*P(not B) = P(not A)*P(not A) = .16 (Since P(B) = P(A), P(not B) = P(not A). Also they are independent events so P that both don't occur is product of P(not A) and P(not B)
P(not A) = .40
So P(A) = .6
This method is faster and innovative.
2: The probability that event B occurs and event A does not is 0.24.
On similar lines,
P(B)*P(not A) = 0.24
x(1 - x) = .24
x could be 0.6 and 1-x would be 0.4 then OR x could be 0.4 and 1-x would be 0.6 then. Hence we don't get a unique value for x
Not sufficient.
Answer (A)