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The probability that event A occurs and event B does not occ

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Carcass
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The probability that event A occurs and event B does not occ [#permalink] New post   07 Aug 2020, 09:32
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The probability that event A occurs and event B does not occur is greater than \(\frac{1}{4}\). The probability that event B occurs and event A does not occur is greater than \(\frac{1}{4}\) . The probability that neither of the events A and B occurs is greater than \(\frac{1}{8}\).


Quantity A
Quantity B
The probability that both of the events A and B occur
\( \frac{1}{4}\)


A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given

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D
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chacinluis
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   08 Aug 2020, 08:41
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Attachment:
vennDiagram.PNG
vennDiagram.PNG [ 8.33 KiB | Viewed 17806 times ]
We have Events A and B, and sample space S

P(A,B complement)>.25
P(A complement, B)>.25
P(A complement, B complement)> .125

We are comparing P(A, B) with 0.25


Look at the Venn diagram posted
First we notice that P(A,B complement)+ P(A complement, B)+ P(A complement, B complement)+ P(A,B)= P(S)= 1

Then we ask ourselves: when will P(A, B) be at its max?
When P(A,B complement), P(A complement, B), and P(A complement, B complement) are its lowest values.

That is when
P(A,B complement)=.25
P(A complement, B)= .25
P(A complement, B complement)= .125

Therefore
P(A,B)<1-P(A,B complement)- P(A complement, B)- P(A complement, B complement)= 1 - .25 - .25 - .125= .375

Therefore P(A,B) could be more or less than .25

Final Answer: D
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   08 Aug 2020, 06:03
Explanation plz?
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   08 Aug 2020, 10:31
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Thank you Sir.

I admire your explanation
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   10 Sep 2020, 12:03
can someone list an actual #, ie .4 or 3/10 for A and for B that will satisfy the original premise?
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   11 Sep 2020, 05:37
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Carcass wrote:
The probability that event A occurs and event B does not occur is greater than \(\frac{1}{4}\). The probability that event B occurs and event A does not occur is greater than \(\frac{1}{4}\) . The probability that neither of the events A and B occurs is greater than \(\frac{1}{8}\).


Quantity A
Quantity B
The probability that both of the events A and B occur
\( \frac{1}{4}\)


We can use a Double Matrix to show all of the probabilities (for more on this watch the video below)

So, one possible case that meets all of the conditions is as follows:
Image
In this case we get:
Quantity A: 0.35
Quantity B: 1/4
Quantity A is greater


Another possible case is as follows:
Image
In this case we get:
Quantity A: 0.22
Quantity B: 1/4
Quantity B is greater


Answer: D

Cheers,
Brent

Double Matrix Method
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   13 Sep 2020, 05:05
Thank you for your great effort! Maybe I am reading the question wrong. But A occurs and B does not occur would be .35 multiplied by .13 which is not greater than 1/4. Also, do the probabilities of occurring and not occurring have to add up to 1?
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   28 Aug 2022, 13:04
P(A, not B) > 1/4
P(not A, B) > 1/4
P(not A, not B) > 1/8
P(A, B) < 1-1/4-1/4-1/8 --> 8/8 - 2/8 - 2/8 - 1/8 --> 3/8

P(A,B)<3/8
so P(A,B) could be greater or less than 1/4=2/8
for example, 2.5/8 (greater than 1/4=2/8) or 1/8 (less than 1/4 = 2/8)
so the answer is D
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The probability that event A occurs and event B does not occ [#permalink] New post   06 Jan 2024, 08:51
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A x B' > 1/4...........i


B x A' > 1/4.........ii

A' x B' > 1/8.........iii



From equation i


B' > 1/4A


From equation ii

A' > 1/4B


sub in equation iii



1/4B x 1/4A > 1/8


Reciprocal reverses inequality sign


16AB < 8

AB < 1/2


Some values will be greater than 1/4 but as you consider other values A becomes less than B

Answer is D.

Adewale Fasipe, GRE, GMAT quant instructor from Nigeria
https://www.teacheron.com/tutor-profile/8tG7?r=8tG7
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   13 Jan 2025, 04:12
adewale223 wrote:
A x B' > 1/4...........i


B x A' > 1/4.........ii

A' x B' > 1/8.........iii



From equation i


B' > 1/4A


From equation ii

A' > 1/4B


sub in equation iii



1/4B x 1/4A > 1/8


Reciprocal reverses inequality sign


16AB < 8

AB < 1/2


Some values will be greater than 1/4 but as you consider other values A becomes less than B

Answer is D.

Adewale Fasipe, GRE, GMAT quant instructor from Nigeria


The problem doesnt state that A and B are independent, so i dont think P(A , B) = P( A) * P(B)
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The probability that event A occurs and event B does not occ [#permalink] New post   31 Jan 2026, 06:24
Im gonna use some obnoxious method for this
Brace yourself

P(A occurs but not B) = 0.25 + x

Cuz it says greater than 0.25

Same for
P(B occurs but not A) = 0.25 + y

and also for

P(Neither B or A occurs) = 0.125 + z

Now yk this

Total = quantity 1 + quantity 2 - both + neither

1 = 0.25+(common part) + x + 0.25 + y + (common part) - (common part) + 0.125 + 7

0.375 = x + y + z + common part

Now common part is your quantity A

0.375 - (x+y+z) = common part

Now you can rewrite it as

0.125 + 0.25 - (x+y+z) = common part

Cancel out 0.25 from both quantity A and B

You'll get quantity A as
0.125-(x+y+z)

And quantity B as 0

Now

x+y+z can be 0.125 or some positive quantity less than 0.125

So its D

kudos is appreciated
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   15 Feb 2026, 16:51
A fairly simplistic numerical evaluation:

We know P(A, no B) > 1/4, P(B, no A) > 1/4, P(no A, no B) > 1/8.

We can check some values, starting with 16 in the denominator to find the sensitivity threshold:

Case 1 (16 in denom):
P(A, no B) = 5/16 (slightly greater than 1/4)
P(B, no A) = 5/16 (slightly greater than 1/4)
P(no A, no B) = 3/16 (slightly greater than 1/8)
(5 + 5 + 3)/16 = 13/16, remainder of 3/16 < 1/4

Case 2 (64 in denom):
P(A, no B) = 17/64 (slightly greater than 1/4)
P(B, no A) = 17/64 (slightly greater than 1/4)
P(no A, no B) = 9/64 (slightly greater than 1/8)
(17 + 17 + 9)/64 = 43/64, remainder of 21/64 > 1/4

Therefore, with contradicting values as the denominator increase, we have the answer of D
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Re: The probability that event A occurs and event B does not occ [#permalink] New post   07 Oct 2026, 06:39
GreenlightTestPrep wrote:
Carcass wrote:
The probability that event A occurs and event B does not occur is greater than \(\frac{1}{4}\). The probability that event B occurs and event A does not occur is greater than \(\frac{1}{4}\) . The probability that neither of the events A and B occurs is greater than \(\frac{1}{8}\).


Quantity A
Quantity B
The probability that both of the events A and B occur
\( \frac{1}{4}\)


We can use a Double Matrix to show all of the probabilities (for more on this watch the video below)

So, one possible case that meets all of the conditions is as follows:
Image
In this case we get:
Quantity A: 0.35
Quantity B: 1/4
Quantity A is greater


Another possible case is as follows:
Image
In this case we get:
Quantity A: 0.22
Quantity B: 1/4
Quantity B is greater


Answer: D

Cheers,
Brent

Double Matrix Method


Thank you for sharing the double matrix method, its pretty helpful
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Re: The probability that event A occurs and event B does not occ [#permalink]
07 Oct 2026, 06:39
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