Joe is among N people in a group, where N 3. If 3 people are randoml

Question

Joe is in a group consisting of N people (N > 4). If 3 people from the group are randomly selected to be on a committee, what is the probability that Joe is selected?

A)  \frac{N^2-2N-6}{(N-2)^2+6}

B)  \frac{2N+3}{N^2-N}

C)  \frac{3}{N}

D)  \frac{N}{(N-1)(N-2)}

E)  \frac{N^2-1}{5N}

Vinit800HBS · Accepted Answer

There are two easy ways to solve this question.

First is what  gmatkarma20  explained in the first post. The benefit of this method is that you can solve by taking any value of N and get the same answer.

The other is to solve in terms of N.
First, total number of cases in which we can select 3 people from N people is: C(n,3) or nC3
N*(N-1)*(N-2)/3*2*1

If Joe has to be in the team, we need to select only 2 more people from the remaining ‘N-1’ people.
This can be done in: (n-1)C2 or C(n-1,2)
(N-1)*(N-2)/2*1

Thus, Probability is:

(N-1)*(N-2)/2*1 
———————— 
N*(N-1)*(N-2)/3*2*1

= 3/N

Option C

GMATPrepNow , Is this approach better?

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gmatkarma20 · Answer

Just plug in a number. Choose N=5, for instance.

# of ways you can form a group of 3: 5C3 = 10

# of ways you can form a group with Joe excluded: 4C3 = 4

Prob(Joe is not selected in the group of 3) = 4/10 = 2/5

Prob(Joe is selected) = 1 - 2/5 = 3/5

C jumps out at you when you plug in N=5 in the options.

So, C.

BrentGMATPrepNow · Answer

Good idea, but when you plug N = 5 into answer choice A, we get 3/5 as well.

A)  \frac{5^2-2(5)-6}{(5-2)^2+6}=\frac{25-10-6}{3^2+6}=\frac{9}{15}=\frac{3}{5}

gmatkarma20 · Answer

Wonderful; down goes my gmat score. Wonderfully deceptive option C.

Vinit800HBS · Answer

gmatkarma20

The extra effort that you have to take when substituting values is to check all options and eliminate the ones that will not end with the desired answer.

Posted from my mobile device

BrentGMATPrepNow · Answer

That's a great approach!

BrentGMATPrepNow · Answer

The correct answer is still C. 
I was just reminding you that, if we use the INPUT-OUTPUT approach, then we need to check  all  of the answer choices.

Cheers,
Brent

Render · Answer

Good, but if you plug N = 10, you get a probability greater than 1.

\frac{10^2-2(10)-6}{(10-2)^2+6}=\frac{100-20-6}{8^2+6}=\frac{74}{70}

Vinit800HBS · Answer

Precisely what I stated to  gmatkarma20

Kindly give kudos  GMATPrepNow  if you liked my approach

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BrentGMATPrepNow · Answer

Great idea! That approach quickly eliminates answer choice A

BrentGMATPrepNow · Answer

Here comes a Kudos . . . . done!!

Vinit800HBS · Answer

Render

But you still need to know that the probability that you are looking for is 3/5.

Let me give you a good idea for substituting values: NEVER PUT THE EXTREME VALUES.

In this, question says N>4. DON’T TAKE N=5 and solve.

Mostly the options are created from the view that student will take N=5 to avoid nasty calculations and will get stuck on multiple options.

Take values that are further away from N= 5. That will avoid falling into these small traps.

Hope that helps.

GMATPrepNow , does this approach sound good?

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BrentGMATPrepNow · Answer

My solution:

If there are N people, then we can select 3 people in NC3 (N choose 3) different ways. 
NC3 =  \frac{(N)(N-1)(N-2)}{3!}  =  \frac{(N)(N-1)(N-2)}{6} 
So, this is our denominator

In how many of the (N)(N-1)(N-2)/6 possible outcomes is Joe chosen? 
Well, once we make Joe one of the selected people, there are N-1 people remaining. 
We can select 2 people from the remaining N-1 people in (N-1)C2 ways
(N-1)C2 =  \frac{(N-1)(N-2)}{2!}  =  \frac{(N-1)(N-2)}{2} 
This is our numerator

So, P(Joe is selected) =  \frac{(N-1)(N-2)}{2}  ÷  \frac{(N)(N-1)(N-2)}{6}

=  \frac{(N-1)(N-2)}{2}  x  \frac{6}{(N)(N-1)(N-2)}

=  \frac{6(N-1)(N-2)}{2(N)(N-1)(N-2)}

=  \frac{6}{2N}

=  \frac{3}{N}

Answer: C

Cheers,
Brent

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BrentGMATPrepNow · Answer

That's sound advice. 
However, I believe  Render  was saying that, once we know the correct answer is A or C, we can quickly eliminate A by testing N = 10
Although it's ideal to eliminate 4 of the 5 answer choices in the first round, Render's approach helps us quickly arrive at the correct answer.

Cheers,
Brent

Mo2men · Answer

Dear  GMATPrepNow  Brent

I'm not able to understand the concept behind your approach. I think in the problem as follows:

Probability to choose Joe =  (choose Joe from from N persons) * (choose 2nd person from N-1 persons) * (choose 3rd person from N-2 persons)  +  (choose 1st person from N persons) *(choose Joe from from N-1 persons) * (choose 3rd person from N-2 persons)  + (choose 1st person from N persons) *(choose 2nd person from N-1 persons) * (choose Joe from from N-2 persons)

Is my approach incorrect?

Thanks in advance

BrentGMATPrepNow · Answer

In your approach, you basically saying that  order matters 
That is, you're saying selecting Joe then Al then Bea is different from Al then Joe then Bea

That approach approach will work as long as you treat the denominator the same way. 
That is, the TOTAL number of ways to select 3 people from N people = (N)(N-1)(N-2)

If you do that, then you should arrive at the correct answer.

Cheers,
Brent

Mo2men · Answer

Dear Brent,

Thanks a lot for your reply. I followed your advice and it yielded the same result.

But I want to understand 2 things in your solution as it is more easier:

1- I used permutation as I think order matters. However, you used combination. why? Does not order matter?

2- In you solution you choose only 2 out of N-1. I do not understand the logic itself. How Joe is included in your solution. I hope you can elaborate.

Thanks in advance

BrentGMATPrepNow · Answer

1) We can go either way with this (i.e., order matters or order does not matter), as long as you use the same construct for numerator and denominator. 
In my approach, I'm assuming order does not matter
For example, selecting Joe then Al then Bea is the SAME as selecting Al then Joe then Bea

2) To count the number of outcomes where Joe is selected to be on the 3-person committee, I first placed Joe on the committee (which means there are N-1 people left to choose from), and then I selected 2 more people to join Joe on the committee. 
So, we are selecting 2 people (from N-1 people) to be on the committee with Joe.

Does that help?

Cheers,
Brent

mgmat4944 · Answer

I don't understand the approach that GMATkarma20 used.

I did it this way:

Probability of picking Joe:

Probability of picking him 1st:

(1/N) * (N-2/N-1) * (N-3/N-2)

since we can have him picked on the 2nd slot or 3rd slot as well we multiple this probability by 3.

we get 3((1/N) * (N-2/N-1) * (N-3/N-2)) = 3(N-1)/N(N-1) =3/N

Joe is among N people in a group, where N > 3. If 3 people are randoml

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Joe is among N people in a group, where N > 3. If 3 people are randoml

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

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745 with Only Self-Prep and GMAT Club

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