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Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
16 May 2012, 07:03

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Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded for first, second, and third place, what is the probability that at least two of the triplets will win a medal?

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
16 May 2012, 07:28

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Expert's post

cnon wrote:

Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded for first, second, and third place, what is the probability that at least two of the triplets will win a medal?

Welcome to GMAT Club. Below is a solution to the question.

The probability that at least two of the triplets will win a medal is the sum of the probability that exactly two of the triplets will win a medal and the probability that all three will win a medal.

The probability that exactly two of the triplets will win a medal is \frac{C^2_3*C^1_6}{C^3_9}=\frac{18}{84}, where C^2_3 is ways to select which two of the triplets will win a medal, C^1_6 is ways to select third medal winner out of the remaining 6 competitors and C^3_9 is total ways to select 3 winners out of 9;

The probability that all three will win a medal is \frac{C^3_3}{C^3_9}=\frac{1}{84};

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
25 Nov 2012, 14:42

From what I understand in the question is that there are 3 teams (triplets) of 3 people each competing together , so how is it possible that only 2 triplets/ teams win medals as there would be a definite 1,2,3 position in case of 3 teams. ?? ---. because the word triathlon implies to me that the teams win not the individuals (teams with cumulative individual lowest score times win) ....

Now if we consider this otherwise that individuals win then :

The probability that at least two of the triplets will win a medal is the sum of the probability that exactly two of the individuals from same team/triplet wins the medal and third medal is won by a individual from one of the remaining 2 teams/triplets + 1 Individual from each team/triplet wins the medal...

P(at least two of the triplets will win a medal is the sum of the probability that exactly two of the individuals from same team/triplet wins the medal and third medal is won by a individual from one of the remaining 2 teams/triplets)=3P2 * 3/9 * 2/8 * 6/7 = 3/7

P(1 Individual from each team/triplet wins the medal)= 3P3 * 3/9 * 3/8 * 3/7 = 9/28

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
26 Nov 2012, 00:55

Expert's post

himanshuhpr wrote:

From what I understand in the question is that there are 3 teams (triplets) of 3 people each competing together , so how is it possible that only 2 triplets/ teams win medals as there would be a definite 1,2,3 position in case of 3 teams. ?? ---. because the word triathlon implies to me that the teams win not the individuals (teams with cumulative individual lowest score times win) ....

Now if we consider this otherwise that individuals win then :

The probability that at least two of the triplets will win a medal is the sum of the probability that exactly two of the individuals from same team/triplet wins the medal and third medal is won by a individual from one of the remaining 2 teams/triplets + 1 Individual from each team/triplet wins the medal...

P(at least two of the triplets will win a medal is the sum of the probability that exactly two of the individuals from same team/triplet wins the medal and third medal is won by a individual from one of the remaining 2 teams/triplets)=3P2 * 3/9 * 2/8 * 6/7 = 3/7

P(1 Individual from each team/triplet wins the medal)= 3P3 * 3/9 * 3/8 * 3/7 = 9/28

3/7 + 9/28 = 3/4 ---- (E)

Pl. help me know where I am going wrong. Thanks.

Your understanding of the question is not correct. Adam, Bruce, and Charlie are not in one team, they compete between each other and 6 other competitors.

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
26 Nov 2012, 06:21

I might be combinatorics challenged... I got that for 2 of the triplets to get award and one other competitor - probability = 3C2 * 6C1 / 9C3 = 3!*6!*3!*6! / 2!*5!*9! = 3*6*3 / 9*8*7 = 3/28

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
30 Nov 2012, 12:55

nanishora wrote:

I might be combinatorics challenged... I got that for 2 of the triplets to get award and one other competitor - probability = 3C2 * 6C1 / 9C3 = 3!*6!*3!*6! / 2!*5!*9! = 3*6*3 / 9*8*7 = 3/28

mCn = m!/(n!*(m-n)!)

How is it that you got 18/84 ???

This calculation is wrong. it will be 3*6*3*2/9*8*7=3/14. Bunuel multiplied it by 6, which comes to 18/84

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
30 Nov 2012, 21:49

Hi Bunuel

Isnt this a permutation problem? I did in the same way as you did except that I used permutation since I thought that the rank and the order in which they are acquired does matter here. Obviously I am wrong since its not the official answer.. What am I missing here??

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 Feb 2013, 15:47

ratinarace wrote:

Hi Bunuel

Isnt this a permutation problem? I did in the same way as you did except that I used permutation since I thought that the rank and the order in which they are acquired does matter here. Obviously I am wrong since its not the official answer.. What am I missing here??

That i was saying to my self tool : the rank and the order does matter here , so we must use permutations. Can someone shed some light about this please ?

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Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 Feb 2013, 17:16

Rock750 wrote:

ratinarace wrote:

Hi Bunuel

Isnt this a permutation problem? I did in the same way as you did except that I used permutation since I thought that the rank and the order in which they are acquired does matter here. Obviously I am wrong since its not the official answer.. What am I missing here??

That i was saying to my self tool : the rank and the order does matter here , so we must use permutations. Can someone shed some light about this please ?

It is a combinatorics question and not a permutation because it asks of two getting a medal (placing either 1st 2nd or 3rd) but does not consider the order.

As a result there are 3 ways in which the podium ( podium is a term used for the top three finishers in a race) can have 2 of the triplets: A and B ; A and C ; B and C. There is only one way in which all three can occupy the podium: A, B, and C. Their finishing position does not count, simply that they made the podium.

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 Feb 2013, 17:37

hitman5532 wrote:

As a result there are 3 ways in which the podium ( podium is a term used for the top three finishers in a race) can have 2 of the triplets: A and B ; A and C ; B and C.

Thanks hitman5532

I think there are more than 3 ways in which the podium can have 2 of the triplets. for instance , there are 3P2 = \frac{3!}{(3-2)!}

which is equals to 6 ways :

A and B A and C B and C B and A C and A C and B

Please correct if i am wrong

_________________

KUDOS is the good manner to help the entire community.

"If you don't change your life, your life will change you"

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 Feb 2013, 19:11

Rock750 wrote:

hitman5532 wrote:

As a result there are 3 ways in which the podium ( podium is a term used for the top three finishers in a race) can have 2 of the triplets: A and B ; A and C ; B and C.

Thanks hitman5532

I think there are more than 3 ways in which the podium can have 2 of the triplets. for instance , there are 3P2 = \frac{3!}{(3-2)!}

which is equals to 6 ways :

A and B A and C B and C B and A C and A C and B

Please correct if i am wrong

It is not a Permutation, it is a combinatoric problem. So the formula is \frac{n!}{k!(n-k)!}

Even without the formula, if you think about what your list shows, you have doubled some selections. A and B is the same as B and A

Think of it this way: Treat the podium as a cup. If you put one A in the cup and one B in the cup, it is no different than one B and one A.

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
09 Feb 2013, 12:33

hitman5532 wrote:

Rock750 wrote:

hitman5532 wrote:

As a result there are 3 ways in which the podium ( podium is a term used for the top three finishers in a race) can have 2 of the triplets: A and B ; A and C ; B and C.

Thanks hitman5532

I think there are more than 3 ways in which the podium can have 2 of the triplets. for instance , there are 3P2 = \frac{3!}{(3-2)!}

which is equals to 6 ways :

A and B A and C B and C B and A C and A C and B

Please correct if i am wrong

It is not a Permutation, it is a combinatoric problem. So the formula is \frac{n!}{k!(n-k)!}

Even without the formula, if you think about what your list shows, you have doubled some selections. A and B is the same as B and A

Think of it this way: Treat the podium as a cup. If you put one A in the cup and one B in the cup, it is no different than one B and one A.

Thanks

I get the point : the order doesn't matter here since all we need here is having at least two of the triplets in the podium ...

Regards

_________________

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Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 May 2013, 19:17

It's been quite some time since this was posted, but I have a quick question about how to solve this problem using fractional probability instead of combinatorics. I tried to solve the problem using the following:

I realize that in order to get to the original answer, I would have had to multiply the initial \frac{6}{84} by 3, and proceed to calculate \frac{19}{84}, but I do not understand why this is the case, since the order of the triplets should not matter.

In another problem from MGMAT's CAT test, I was able to solve for the desired probability using fractional probabilities only:

"Bill has a small deck of 12 playing cards made up of only 2 suits of 6 cards each. Each of the 6 cards within a suit has a different value from 1 to 6; thus, for each value from 1 to 6, there are two cards in the deck with that value. Bill likes to play a game in which he shuffles the deck, turns over 4 cards, and looks for pairs of cards that have the same value. What is the chance that Bill finds at least one pair of cards that have the same value?"

Of course, this problem could have also been solved using combinatorics, but I feel the fractional approach would probably be quicker in a live scenario. However, over the course of my review, I've found the fractional method to be unreliable, whereas combinatorics are typically universally applicable (though they can take significantly longer in some instances).

I am curious as to why the solution for the second problem involving the playing cards worked whereas my solution to the original triplet question did not. I am not really sure why I have to multiply that initial \frac{6}{84} by 3 to get the right answer, and I would like to have a concrete understanding of this issue to help me solve similar problems.

Last edited by wfa207 on 08 May 2013, 04:35, edited 1 time in total.

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 May 2013, 19:50

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Expert's post

Hey wfa,

You're right about order not mattering but your math suggests you're missing a fundamental counting concept.

P(2 triplets) is not simply equal to 3/9*2/8*6/7 as this would suggest a sequence where order matters. Rather, the math is (number of successful outcomes)/(total number of outcomes)

Total # of ways you could pick 2 of the triplets among the medallists is C(2,3), AND total # ways you could pick 1 non-triplet among the medallists is C(1,6), that's 3*6= 18

Total # outcomes is the # of ways you could pick ANYONE to be medallists, so C(6,9) = 84

P(2 triplets) = 18/84

I know u we're off by a factor of 3, and you think it's an order vs no-order issue, but really, you were attempting sequential counting, but should have been using combinatorials

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
07 May 2013, 20:11

DD, thanks for the quick response.

I'm still a little fuzzy on why \frac{3}{9} \times \frac{2}{8} \times \frac{6}{7} from the triplets problem implies that order matters whereas the \frac{12}{12} \times \frac{10}{11} \times \frac{8}{10} \times \frac{6}{9} from the card pairs problem does not.

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
18 Aug 2013, 11:22

1

This post received KUDOS

cnon wrote:

Triplets Adam, Bruce, and Charlie enter a triathlon. If there are 9 competitors in the triathlon, and medals are awarded for first, second, and third place, what is the probability that at least two of the triplets will win a medal?

Re: Triplets Adam, Bruce, and Charlie enter a triathlon. If [#permalink]
19 Aug 2013, 01:55

Rock750 wrote:

hitman5532 wrote:

As a result there are 3 ways in which the podium ( podium is a term used for the top three finishers in a race) can have 2 of the triplets: A and B ; A and C ; B and C.

Thanks hitman5532

I think there are more than 3 ways in which the podium can have 2 of the triplets. for instance , there are 3P2 = \frac{3!}{(3-2)!}

which is equals to 6 ways :

A and B A and C B and C B and A C and A C and B

Please correct if i am wrong

In case the order does matter we would still get the same answer i believe.. Experts please correct me if i am wrong...

No of ways of choosing 3 ppl out of 9 = 3C9=84

No of ways those three ppl can be arranged on the podium = 6

So hence total number of options = 504

No of ways A,B and C can be arranged on the podium = 6

No of ways you can choose two ppl from A, B, C = 3 No of ways you can chose the 3rd person from 6 = 6 No of ways you can arrange 3 ppl on the podium = 6

So total number of ways = 6 x 6 x3 = 108

So probability of at least two on the podium = 108+6/504 = 19/84

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