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

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Math Expert
Joined: 02 Sep 2009
Posts: 64242
The events A and B are independent. The probability that event A occur  [#permalink]

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10 Jul 2016, 22:58
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55% (hard)

Question Stats:

50% (01:05) correct 50% (01:08) wrong based on 228 sessions

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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

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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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10 Jul 2016, 23:29
2
Bunuel wrote:
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

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

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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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11 Jul 2016, 02:37
1
2
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
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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11 Jul 2016, 08:43
1
prashant212 wrote:
Bunuel wrote:
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

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

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
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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13 Jul 2016, 07:45
P does and q does not or q does and p does not
p(1-q)+q(1-p),
p+q-2pq
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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13 Jul 2016, 20:12
Bunuel wrote:
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

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?
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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10 Jun 2018, 08:51
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)
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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10 Jun 2018, 08:52
Anigr16 wrote:
Bunuel wrote:
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

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?

Yes
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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12 Aug 2019, 10:50
Bunuel wrote:
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

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.

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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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14 Mar 2020, 11:08

"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

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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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14 Mar 2020, 23:13
1
jabhatta@umail.iu.edu wrote:

"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

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.
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Re: The events A and B are independent. The probability that event A occur  [#permalink]

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15 Mar 2020, 02:46
Bunuel wrote:
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

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
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Re: The events A and B are independent. The probability that event A occur   [#permalink] 15 Mar 2020, 02:46