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What is the probability P that A and B are both

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COolguy101
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What is the probability P that A and B are both [#permalink] New post   Updated on: 29 Jun 2021, 09:36
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What is the probability P that A and B are both selected when you select 40 people out of a group of 100 (A and B included)?

Quantity A
Quantity B
P
26/165



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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C

Originally posted by COolguy101 on 27 Jun 2021, 06:52.
Last edited by KarunMendiratta on 29 Jun 2021, 09:36, edited 2 times in total.
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What is the probability P that A and B are both [#permalink] New post   29 Jun 2021, 09:44
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kumarneupane4344 wrote:
What is the probability P that A and B are both selected when you select 40 people out of a group of 100 (A and B included)?

Quantity A
Quantity B
P
26/165



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.


We already have A and B, so we need to pick 38 more people from 100 - 2 = 98 people
This can be done in \(^{98}C_{38}\)
Total ways = \(^{100}C_{40}\)

P = \(\frac{^{98}C_{38}}{^{100}C_{40}}\)

P = \(\frac{98!}{(38!)(60!)} \frac{(40!)(60!)}{100!}\)

P = \(\frac{(98!)(40!)}{(38!)(100!)} = \frac{(40)(39)}{(100)(99)} = \frac{26}{165}\)

Hence, option C
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COolguy101
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Re: What is the probability P that A and B are both [#permalink] New post   02 Jul 2021, 07:11
KarunMendiratta wrote:
kumarneupane4344 wrote:
What is the probability P that A and B are both selected when you select 40 people out of a group of 100 (A and B included)?

Quantity A
Quantity B
P
26/165



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.


We already have A and B, so we need to pick 38 more people from 100 - 2 = 98 people
This can be done in \(^{98}C_{38}\)
Total ways = \(^{100}C_{40}\)

P = \(\frac{^{98}C_{38}}{^{100}C_{40}}\)

P = \(\frac{98!}{(38!)(60!)} \frac{(40!)(60!)}{100!}\)

P = \(\frac{(98!)(40!)}{(38!)(100!)} = \frac{(40)(39)}{(100)(99)} = \frac{26}{165}\)

Hence, option C

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Re: What is the probability P that A and B are both [#permalink] New post   30 Aug 2022, 06:50
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Solution -

We can use the combination concept here without using the formula.
Combinations are easy-going and the order doesn’t matter.

In Quantity A, they are asking for Probability.
Probability of an Event (P) = Number of Desired Outcome / Total Number of Possible Outcome

First we will consider the Number of Desired Outcome-

So, we are selecting both A and B i.e., two people. Out of 40 people, we are picking A and B.
We have 40 choices for the first person (say A), and, for the second person (i.e., B) we have 39 choices.
We are choosing two people but we are not mentioning the order. Thus, any way you choose them, it doesn't make any difference.

Now, what should we do to get rid of repetitions? - To get rid of the repetition of similar combinations; we'll divide the result by a factorial of 2.

Why 2!? - Because the number of elements in the numerator is two, as shown below:

So, the first part will be - (40*39)/2!

Now, looking into the next part i.e., where we will consider the Total Number of Possible Outcome-

Here, also we are selecting both A and B i.e., two people, but here, out of 100 people, we are picking A and B.
We have 100 choices for the first person (say A), and, for the second person (i.e., B) we have 99 choices.

We will use the same logic, and thus, the second part will be - (100*99)/2!

Finally we use the probability formula, i.e., Probability of an Event (P) = Number of Desired Outcome / Total Number of Possible Outcome

=> [(40*39)/2!] / [(100*99)/2!]

By reducing the above fraction, we get 26/165

Thus, Quantity A is 26/ 165

Quantity B is also 26 / 165

Thus, Quantity A = Quantity B

The Correct Answer Choice is C
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What is the probability P that A and B are both [#permalink] New post   30 Aug 2022, 10:44
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KarunMendiratta used the Reverse Probability Method.
SeaOyster1 used the Slot Method.

Nice to see both methods side by side.
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Re: What is the probability P that A and B are both [#permalink] New post   29 Feb 2024, 23:36
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Re: What is the probability P that A and B are both [#permalink]
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