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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 [#permalink]

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New post Updated on: 28 Feb 2014, 07:22
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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.

Spoiler: ::
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.
Spoiler: OA

Originally posted by goodyear2013 on 27 Feb 2014, 12:18.
Last edited by goodyear2013 on 28 Feb 2014, 07:22, edited 1 time in total.
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Re: Events A and B are independent and have equal probabilities [#permalink]

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New post 27 Feb 2014, 21:59
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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.

Spoiler: ::
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)
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Events A and B are independent and have equal probabilities [#permalink]

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New post 19 Jul 2015, 17:18
The Statement (2) is not sufficient but I think the explanation is wrong. You can't multiply that way because those probabilities have intersection (not independent).
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Re: Events A and B are independent and have equal probabilities [#permalink]

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New post 19 Jul 2015, 17:29
Andrake26 wrote:
The Statement (2) is not sufficient but I think the explanation is wrong. You can't multiply that way because those probabilities have intersection (not independent).


The question does mention that the two events, A and B are independent. Thus the intersection is not there.

This means, P(A and B) = P (A) * P(B) and P(A or B) = P(A) + P (B).

The explanation given by VeritasPrepKarishma is correct.
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Re: Events A and B are independent and have equal probabilities [#permalink]

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New post 05 Aug 2015, 04:14
VeritasPrepKarishma wrote:
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.

Spoiler: ::
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)



Karishma I think this is the Formula -

For Mutually Exclusive events : Those events which will have nothing in common between them.
2 different sample space.
P (A or B ) = P(A) + p(B)
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Events A and B are independent and have equal probabilities [#permalink]

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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.

Spoiler: ::
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.


Hi goodyear2013, statement (1) implies that probability of occurrence of at least one of the two events is 0.84. Hence, the given probability (0.84) includes 3 possibilities:
1. A happens and B does not
2. B happens and A does not
3. A and B both happen

In the first case, the probability can be written as P(A)*[1 - P(B)]. Similarly, in the second case, probability can be calculated as \([1 - P(A)]*P(B)\); and in the third case, probability is P(A)*P(B). This will give the equation as: \(2p(1 - p) + {p}^{2} = 0.84\) and value of p (as Karishma explained) will be 3/5.

Alternate Method:
If we consider all the cases there can be when 2 events do or do not occur, then they are as follows:
1. both A and B occur
2. A occurs and B does not
3. B occurs and A does not
4. both A and B do not occur
Hence, addition of the probabilities of the above four cases should be 1. Since the first statement considers the combined probability of the first three cases, thus (1 - 0.84) should be the probability of fourth case. It implies that 0.16 = (1 - p)*(1 - p).

Hope this explanation helps.
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Re: Events A and B are independent and have equal probabilities [#permalink]

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honchos wrote:

Karishma I think this is the Formula -

For Mutually Exclusive events : Those events which will have nothing in common between them.
2 different sample space.
P (A or B ) = P(A) + p(B)


Independent events are not the same as mutually exclusive events.

When events are independent, they can both occur at the same time. For example:
Team A will win the match tomorrow.
It will rain tomorrow.
These events are independent. One does not depend on the other but they both can occur tomorrow.
P(A or B) = P(A) + P(B) - P(A)*P(B)

It will rain less than 1 mm tomorrow.
It will rain more than 3 mmm tomorrow.
These two events are mutually exclusive and cannot happen at the same time.
P(A or B) = P(A) + P(B)
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Re: Events A and B are independent and have equal probabilities [#permalink]

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New post 14 Apr 2017, 13:13
the problem is just another a simple equation
Re: Events A and B are independent and have equal probabilities   [#permalink] 14 Apr 2017, 13:13
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