Events A and B are independent and have equal probabilities : Data Sufficiency (DS)

Andrake26

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

19 Jul 2015, 17:18

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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]

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]

05 Aug 2015, 04:14

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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.

Show 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]

05 Aug 2015, 05:04

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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.

Show 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.

B

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

14 Apr 2017, 13:13

the problem is just another a simple equation

L

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

20 Jul 2018, 22:53

Expert Reply

Since A and B have equal probabilities of occurring (let’s call that probability p), they also have equal probabilities of not occurring. The probability that neither of the events occurs is .16 (the complementary probability of .84, or 1 - .84). Therefore, the probability of A not occurring and the probability of B not occurring is .16.
That means that (not p)(not p) = .16, or \((not p)^2\) = .16.

Take the square root of both sides to find that the probability of an event not occurring is .4. If we know the probability that an event will not occur, then we know the probability that it will occur (in this case, it is .6, or 60%).
So Statement (1) is sufficient.

Statement (2): The probability that event A does not occur is simply 1 – p, so Statement (2) gives us p(1 – p) = .24. This quadratic becomes \(p^2 - p + .24 = 0\), and factoring for two numbers that multiply to .24 and add to -1 (the implicit coefficient in front of p) gives us two possible values for p: .6 and .4.
So Statement (2) alone is insufficient.

The correct answer is therefore (A).

B

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

31 Aug 2020, 18:48

VeritasKarishma 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.

Show 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)

Hi,

But if you consider the case in which both A and B are disjoint, then statement 1 by itself won't be sufficient.

So I think the answer is E.

Thanks

G

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

13 Feb 2021, 21:16

VeritasKarishma 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.

Show 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)

I think this is where I got confused with statement 1. I interpreted "The probability that at least one of events A and B" as the following probability:

P(A or B or both) rather than ...

P(A or B) = P(A) + P(B) - P(A and B)

VeritasKarishma can you clarify?

L

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

16 Feb 2021, 00:08

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CEdward wrote:

VeritasKarishma 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.

Show 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)

I think this is where I got confused with statement 1. I interpreted "The probability that at least one of events A and B" as the following probability:

P(A or B or both) rather than ...

P(A or B) = P(A) + P(B) - P(A and B)

VeritasKarishma can you clarify?

P(A or B) is the same as P(A or B or Both). When we say P(A or B), we mean to say 'probability that A happens or B happens or both happen' i.e. at least one of them does take place and both could also take place.

P(A or B) = P(A) + P(B) - P(Both)
includes the case where Both take place. We subtract P(Both) above because it is counted twice, once in P(A) and then in P(B). But we need to count it only once.

L

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

20 Jun 2021, 06:03

VeritasKarishma 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.

Show 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)

VeritasKarishma how did you get this x(1 - x) = .24 from this P(B)*P(not A) = 0.24

L

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

20 Jun 2021, 22:47

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Expert Reply

dave13 wrote:

VeritasKarishma 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.

Show 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)

VeritasKarishma how did you get this x(1 - x) = .24 from this P(B)*P(not A) = 0.24

Events A and B are independent and have equal probabilities of occurring.

So say P(A) = x
P(not A) = 1 - x (Complementary event of A)

P(B) = x (because not events have equal probability of occurring)
P(not B) = 1 - x (Complementary event of B)

Given:
P(B)*P(not A) = 0.24
x * (1 - x) = 0.24

L

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

21 Jun 2021, 02:04

Quote:

Events A and B are independent and have equal probabilities of occurring.

So say P(A) = x
P(not A) = 1 - x (Complementary event of A)

P(B) = x (because not events have equal probability of occurring)
P(not B) = 1 - x (Complementary event of B)

Given:
P(B)*P(not A) = 0.24
x * (1 - x) = 0.24

VeritasKarishma thanks, just one question:

Probability of A and B are equal, say A= 1/4 and B =1/4

then as per your explanation i get 1/4 -1

also "1" in probability of indepedent events implies this p(a)+p(b)-both+neither isnt it so ?

now whats wrong with my reasoning

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IanStewart

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

21 Jun 2021, 02:55

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dave13 wrote:

Probability of A and B are equal, say A= 1/4 and B =1/4

then as per your explanation i get 1/4 -1

also "1" in probability of indepedent events implies this p(a)+p(b)-both+neither isnt it so ?

As Karishma said, if the probability A happens is x, the probability A does not happen is 1-x. If x = 1/4, then the probability A does not happen is 1 - (1/4) = 3/4 (it is not 1/4 - 1).

The formula you write out is just a Venn diagram formula, so it will always be correct provided you have a Venn situation, though I've never used it to answer a GMAT probability question. If as you suggest P(A) = 1/4, then P(B) = 1/4 (the probabilities are equal in this question), P(both) = (1/4)(1/4) = 1/16, and P(neither) = (3/4)(3/4) = 9/16. If you now plug into your formula, you will indeed get 1 as the result: 1/4 + 1/4 - 1/16 + 9/16 = 1.

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

05 Jan 2024, 01:55

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

05 Jan 2024, 01:55

GMAT Data Sufficiency (DS) Questions

Events A and B are independent and have equal probabilities