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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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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 13127 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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Farina
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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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rydercourtney1
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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
1
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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The probability that event A occurs and event B does not occ [#permalink]
06 Jan 2024, 08:51
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