The events A and B are independent. The probability that event A occur

Question

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 and B occurs?

A. p − 2pq
B. q − pq
C. p + q − 2pq
D. p + q − pq
E. p + q

A. p − 2pq
B. q − pq
C. p + q − 2pq
D. p + q − pq
E. p + q

Bunuel · Accepted Answer

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 and B occurs?

A. p − 2pq
B. q − pq
C. p + q − 2pq
D. p + q − pq
E. p + q

A. p − 2pq 
B. q − pq 
C. p + q − 2pq 
D. p + q − pq 
E. p + q

KAPLAN OFFICIAL EXPLANATION

Let's say that P(A) is the probability that event A occurs, P(B) is the probability that event B occurs, P(not A) is the probability that event A does not occur, P(not B) is the probability that event B does not occur, P(A and not B) is the probability that event A occurs and event B does not occur, and P(not A and B) is the probability that event A does not occur and event B occurs. Then P(A) = p and P(B) = q. The probability that an event does not occur is equal to 1 minus the probability that the event occurs. So P(not A) = 1 − p and P(not B) = 1 − q.

The probability that exactly one of the events A and B occur is P(A and not B) + P(not A and B). The events A and B are independent, so P(A and not B) = P(A)P(not B) = p(1 − q) = p − pq. Again, the events A and B are independent, so P(not A and B) = P(not A)P(B) = (1 − p)q = q − pq.

Thus, P(A and not B) = p − pq and P(not A and B) = q − pq.

So the probability that exactly one of the events A and B occurs is P(A and not B) + P(not A and B) = p − pq + q − pq = p + q − 2pq.

Choice (C) is correct.

iPinnacle · Answer

shouldn't this be as simple as below.

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

so
p*(1-q)+q*(1-p)
=p-pq+q-pq
=p+q-2pq

KapTeacherEli · Answer

iPinnacle's got a great explanation for this particular problem. However, there is a basic principle here that I wanted to take a minute to explain.

Combined probability is a lot like overlapping sets in set theory. Several previous posters tried to figure out the odds of at least one of A and B by multiplying their probabilites, p and q. However, if we are counting the odds of A, and independently counting the odds of A,  that means any time A and B both happen, we've counted twice!  Thus, the odds of at least one of A and B occuring is p + q - pq--subtracting the overlap to offset the double counting.

Then, since overlaps are invalid in this particular problem, we subtract the overlap, pq,  again , getting the same place iPinnacle did.

prashant212 · Answer

Probability of A occuring = p , Probability of A NOT occuring = 1-p
Probability of B occuring = q , Probability of B not occurring = 1-q

Probability of atleast of one of A and B occuring = 1-(1-p)(1-q) =  p+q -pq 
 
D

gaurav2187 · Answer

I ll go with C

Probability of A happening and B not happrning- p*(1-q)
Probability of B happening and A not happening- q*(1-p)

Exactly 1 of them happening- p*(1-q)+ q*(1-p) = p+q-2pq

sourcex · Answer

This is not right. When you consider atleast one, it also means you are considering the possibility of both events occuring, which should be excluded as per the question.

So correct would be A occurs, B doesn't occur and B occurs, A doesn't occur = p(1-q) + q(1-p) = p+q-2pq

CounterSniper · Answer

P does  and  q does not  or  q does  and  p does not 
p(1-q)+q(1-p),
p+q-2pq

Anigr16 · Answer

Just to confirm, if we had to find the probability that one of the events A and B occurs (omitted the highlighted text)-
it would have been:
P(A or B) = P(A) + P(B) - P(A and B)
P(A or B) = p+ q- pq

Is this correct?

praneethuddagiri · Answer

Probability of Exactly one of A or B occurring if I write it in Union form:
P(Exactly one occurring) = P(a U b) - P(a n b) = p(a) + p(b) - 2(p(a n b)) = p + q - 2pq(since a and b are independent)

praneethuddagiri · Answer

Yes

ScottTargetTestPrep · Answer

Because events A and B are independent, the probability that both A and B occur is the product of their individual probabilities, so we have P(A and B) = pq.

Now, note that the event “A only” means that the part of A that includes both A and B happening is not part of this event. Thus, we have:

P(A only) = P(A) - P(A and B) = p - pq.

Similarly, for the event “B only” we see that

P(B only) = P(B) - P(A and B) = q - pq.

Thus, the probability that exactly one event happens (i.e., A only or B only) is equal to

p - pq + q - pq = p + q - 2pq.

Answer: C

jabhatta2 · Answer

"Because events A and B are independent, the probability that both A and B occur is the product of their individual probabilities"

Hi Experts  chetan2u ,  VeritasKarishma ,  Bunuel ,  generis ,  MahmoudFawzy

Why is that when two events are independent, the product of BOTH occurring at the same is a cross multiplication

I thought logically, the product of BOTH events happening at the same time when independent would be zero

Please assist in this logic

KarishmaB · Answer

Independent events means they are not related to each other. They are not necessarily mutually exclusive. 
e.g. the probability that it will rain in the day and the probability of getting a surprise test in school are independent. One doesn't depend on the other. It is possible that both will happen, only one will happen or neither will happen.

Say probability of rain is 1/5 
Probability of a surprise test is 1/4 
Probability that on any given day, both will happen = 1/5 * 1/4 = 1/20

Mutually exclusive events are those in which if one happens the other doesn't happen. 
e.g. "a friend will visit me today" and "no one will visit me today" are mutually exclusive. If one happens, other cannot happen.

Kinshook · Answer

Given: 
1. The events A and B are independent. 
2. The probability that event A occurs is p and the probability that event B occurs is q.

Asked: Which of the following is equal to the probability that exactly one of the events A and B occurs?

Probability that even A & even B occurs = pq

Probability that exactly one the events A and B occurs = A not B + B not A = p (1-q) + q (1-p) = p + q - 2pq

IMO C

NEYR0N · Answer

what if we have 3 probabilities ? how do we find exactly 1 ?

Krunaal · Answer

Probability that exactly one of the events A, B, and C occurs = P(A occurs)*P(B does not occur)*P(C does not occur) + P(B occurs)*P(A does not occur)*P(C does not occur) + P(C occurs)*P(A does not occur)*P(B does not occur)

NEYR0N · Answer

Krunaal  can this be applied in 3 overlapping sets where some unions are given but the question asks for EXACTLY 1 ?

The events A and B are independent. The probability that event A occur

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The events A and B are independent. The probability that event A occur

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