Probability doubt.

Question

Hey can somebody please help me with this question.
I faced it while giving the Kaplan practice test.

Q: The events A and B are independent. The probability that event A occurs is p and the probability that event B occurs is q. Which of the following is equal to the probability that exactly one of the events A or B occur.

p-2pq
p-pq
p+q-pq
p+q-2pq
p+q

Now, what I did was use the formula : p(that event A occurs) + p(that event B occurs) - p(both events occur)
which gave p + q - pq

But the answer is actually p + q - 2pq.

Thanks.

Anurag@Gurome · Answer

Aishwarya1204 wrote:Hey can somebody please help me with this question.
I faced it while giving the Kaplan practice test.

Q: The events A and B are independent. The probability that event A occurs is p and the probability that event B occurs is q. Which of the following is equal to the probability that exactly one of the events A or B occur.

p-2pq
p-pq
p+q-pq
p+q-2pq
p+q

Now, what I did was use the formula : p(that event A occurs) + p(that event B occurs) - p(both events occur)
which gave p + q - pq

But the answer is actually p + q - 2pq.

Thanks.

Probability that exactly one of the events A and B occurs means if A occurs then B does not occur or if B occurs then A does not occur.

Therefore, p * (1 - q) + q * (1 - p) = p - pq + q - pq
= p + q - 2pq

The correct answer is D.

Anurag Mairal, Ph.D., MBA
GMAT Expert, Admissions and Career Guidance
Gurome, Inc.
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Aishwarya1204 · Answer

Anurag@Gurome wrote:Aishwarya1204 wrote:Hey can somebody please help me with this question. I faced it while giving the Kaplan practice test. Q: The events A and B are independent. The probability that event A occurs is p and the probability that event B occurs is q. Which of the following is equal to the probability that exactly one of the events A or B occur. p-2pq p-pq p+q-pq p+q-2pq p+q Now, what I did was use the formula : p(that event A occurs) + p(that event B occurs) - p(both events occur) which gave p + q - pq But the answer is actually p + q - 2pq. Thanks. Probability that exactly one of the events A and B occurs means if A occurs then B does not occur or if B occurs then A does not occur. Therefore, p * (1 - q) + q * (1 - p) = p - pq + q - pq = p + q - 2pq The correct answer is D. Thank You so much ! But can you tell me what the main difference is between the two. The formula ideally means that exactly one event occurs as well right?

Anurag@Gurome · Answer

Aishwarya1204 wrote: Thank You so much !
But can you tell me what the main difference is between the two. The formula ideally means that exactly one event occurs as well right?

When we count the odds of A, and independently counting the odds of A means any time A and B both happen, which means we are counting two times.
Therefore, the odds of at least one of A and B occurring is p + q - pq - subtracting the overlap.

In this question, we subtract the overlap, pq, again. I hope it helps.

Brent@GMATPrepNow · Answer

Aishwarya1204 wrote: Q: The events A and B are independent. The probability that event A occurs is p and the probability that event B occurs is q. Which of the following is equal to the probability that exactly one of the events A or B occur.
p-2pq
p-pq
p+q-pq
p+q-2pq
p+q

If events A and B are independent, then P(A and B) = P(A) x P(B)

Now let's reword the probability. To do this, I always ask, "What needs to occur here to achieve the stated outcome?"

We get: P(A only OR B only) = P(A and not B _ OR _B and not A)
= P(A and not B) + P(B and not A)
= [P(A) x P(not B)] + [P(B) x P(not A)]

Aside: If P(A) = p, then P(not A) = 1 - p
Similarly, P(B) = q, then P(not B) = 1 - q

Now replace the given probabilities to get:
= [p x (1-q)] + [q x (1-p)]
Simplify to get: [p - pq] + [q - pq]
= p + q - 2pq
= D

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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