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SVP  Joined: 21 Jan 2007
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Location: New York City
A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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Difficulty:   15% (low)

Question Stats: 80% (01:50) correct 20% (02:15) wrong based on 453 sessions

HideShow timer Statistics A basket contains 3 blue, 3 red and 3 yellow marbles. If 3 marbles are extracted from the basket at random, what is the probability that a marble of each color is among the extracted?

A. 2/21
B. 3/25
C. 1/6
D. 9/28
E. 11/24

Originally posted by bmwhype2 on 16 Nov 2007, 07:46.
Last edited by Bunuel on 17 Jul 2012, 02:43, edited 1 time in total.
Edited the question and added the OA.
Math Expert V
Joined: 02 Sep 2009
Posts: 55804
A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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9
2
malikshilpa wrote:
Is ' picking of 3 marbles at random' same as 'picking 3 marbles one by on without replacement' ? Although looking at the solution these two scenarios appear to be similar. However, when 3 marbles are picked up together (in one shot) then should the problem not be treated differently. Please clarify .

Mathematically the probability of picking 3 marbles simultaneously, or picking them one at a time (without replacement) is the same.

A basket contains 3 blue, 3 red and 3 yellow marbles. If 3 marbles are extracted from the basket at random, what is the probability that a marble of each color is among the extracted?
A. 2/21
B. 3/25
C. 1/6
D. 9/28
E. 11/24

We need to find the probability of BRY (a marble of each color).

Probability approach:

$$P(BRY)=\frac{3}{9}*\frac{3}{8}*\frac{3}{7}*3!=\frac{9}{28}$$, we are multiplying by 3! since BRY scenario can occur in several different ways: BRY, RYB, RBY (# of permutations of 3 distinct letters BRY is 3!).

Combination approach:

$$P(BRY)=\frac{C^1_3*C^1_3*C^1_3}{C^3_9}=\frac{9}{28}$$.

Hope it's clear.
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VP  Joined: 08 Jun 2005
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9/9*6/8*3/7 = 9/28 Manager  Joined: 26 Sep 2007
Posts: 57

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1
Probability of getting each color marble = (3C1*3C1*3C1)/9C3
= (3*3*3)/84
= 27/84 = 9/28
Director  Joined: 09 Aug 2006
Posts: 693

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KillerSquirrel wrote:
9/9*6/8*3/7 = 9/28 KS, can you please explain how you get 3/7?

I'm assuming you did the following:
The 1st chosen can be any color and it doesn't really matter so let's say the first chosen is Blue.

The probability that the 2nd chosen is not blue = 1-2/8 = 6/8

Now what?? How do I proceed? Thanks!
VP  Joined: 08 Jun 2005
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GK_Gmat wrote:
KillerSquirrel wrote:
9/9*6/8*3/7 = 9/28 KS, can you please explain how you get 3/7?

I'm assuming you did the following:
The 1st chosen can be any color and it doesn't really matter so let's say the first chosen is Blue.

The probability that the 2nd chosen is not blue = 1-2/8 = 6/8

Now what?? How do I proceed? Thanks!

You are right about the first marble , you can choose any marble out of nine = 9/9 = 1 (nine out of nine).

For the second marble you are also correct - since you removed one marble from the basket you now have eight left but you can't choose the color already been choosen so 6/8 (six out of eight - three for one color and three for the other color).

For the third marble the logic is the same - you can choose one of the three marbles from the color not already choosen and you have seven mables to choose from meaning 3/7 (three out seven).

All of this is true only when the marbles taken out are not returned to the basket - if they were the probability then was ?? - can you solve this ? SVP  Joined: 29 Aug 2007
Posts: 2258
Re: Probability - Marbles of each color  [#permalink]

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bmwhype2 wrote:
A basket contains 3 blue, 3 red and 3 yellow marbles. If 3 marbles are extracted from the basket at random, what is the probability that a marble of each color is among the extracted?

2/21
3/25
1/6
9/28
11/24

alternatively:

= (3c1 x 3c1 x 3c1)/9c3
= 9/28
SVP  Joined: 21 Jan 2007
Posts: 2477
Location: New York City

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sevenplus wrote:
Probability of getting each color marble = (3C1*3C1*3C1)/9C3
= (3*3*3)/84
= 27/84 = 9/28

yep. most intuitive approach.

desired/total

OA is D
Director  Joined: 01 May 2007
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For me it was easier to just write it out.

I need 3 different colors so, we'll say I wanted a red, blue, yellow.

So chance of picking a red = 3/9

On 2nd pick chance of picking a blue = 3/8

On 3rd pick chance of picking a yellow = 3/7.

Then I listed out the different 3 ball combos I could have.

BRY
BYR
RBY
RYB
YRB
YBR

So 6 different combos. So the prob of pick r,b,y is 3/9 * 3/8 * 3/7 = 3/56

Multiply that by 6 and you have 9/28 prob of picking three different colored balls.
Manager  Joined: 27 Oct 2008
Posts: 164
Re: Probability - Marbles of each color  [#permalink]

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A basket contains 3 blue, 3 red and 3 yellow marbles.
If 3 marbles are extracted from the basket at random, what is the probability that a marble of each color is among the extracted?

2/21
3/25
1/6
9/28
11/24

Soln:
= 3C1 * 3C1 * 3C1/9C3
= 27/(9 * 8 * 7/6)
= 27/84
= 9/28
Senior Manager  Joined: 08 Nov 2010
Posts: 316
Re: Probability - Marbles of each color  [#permalink]

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Hey guys,

I understand the solution, but i need someone to explain me why the following is not working:

total groups of 3 - 9C3

total options of 3 different color:
abc
acb
bac
bca
cab
cba

so - 6 options to get 3 different colors.

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Manager  Joined: 25 May 2011
Posts: 111
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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1
2
We can choose 3 balls from 9 balls. Total possibilities: $$C^9_3$$
We should choose 1 ball from each color. Favorite possibilities: $$C^3_1*C^3_1*C^3_1$$

$$p=\frac{C^3_1*C^3_1*C^3_1}{C^9_3}=\frac{9}{28}$$
D
Intern  Joined: 04 Jun 2012
Posts: 2
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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Is ' picking of 3 marbles at random' same as 'picking 3 marbles one by on without replacement' ? Although looking at the solution these two scenarios appear to be similar. However, when 3 marbles are picked up together (in one shot) then should the problem not be treated differently. Please clarify .
Intern  Joined: 29 Mar 2014
Posts: 9
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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Just wanted to clarify here, as I feel like I'm looking at this wrong
n should be 3 (because this is the total set)
r should be 1 (this is the number of objects being extracted)

so shouldn't the combination formula be C 3 over 1? Instead of C 1 over 3?
Math Expert V
Joined: 02 Sep 2009
Posts: 55804
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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TheBookie wrote:
Just wanted to clarify here, as I feel like I'm looking at this wrong
n should be 3 (because this is the total set)
r should be 1 (this is the number of objects being extracted)

so shouldn't the combination formula be C 3 over 1? Instead of C 1 over 3?

$$C^1_3$$, $$C^3_1$$, 3C1, mean the same thing choosing 1 out of 3, just different ways to write. Can it be choosing 3 out of 1???
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Posts: 9
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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Bunuel wrote:
TheBookie wrote:
Just wanted to clarify here, as I feel like I'm looking at this wrong
n should be 3 (because this is the total set)
r should be 1 (this is the number of objects being extracted)

so shouldn't the combination formula be C 3 over 1? Instead of C 1 over 3?

$$C^1_3$$, $$C^3_1$$, 3C1, mean the same thing choosing 1 out of 3, just different ways to write. Can it be choosing 3 out of 1???

Agreed, but for the sake of consistency, I see the combination formula written as C n over r
http://www.mathwords.com/c/combination_formula.htm

Thanks
Manager  Joined: 15 Aug 2013
Posts: 240
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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Bunuel wrote:
malikshilpa wrote:
Is ' picking of 3 marbles at random' same as 'picking 3 marbles one by on without replacement' ? Although looking at the solution these two scenarios appear to be similar. However, when 3 marbles are picked up together (in one shot) then should the problem not be treated differently. Please clarify .

Mathematically the probability of picking 3 marbles simultaneously, or picking them one at a time (without replacement) is the same.

A basket contains 3 blue, 3 red and 3 yellow marbles. If 3 marbles are extracted from the basket at random, what is the probability that a marble of each color is among the extracted?
A. 2/21
B. 3/25
C. 1/6
D. 9/28
E. 11/24

We need to find the probability of BRY (a marble of each color).

Probability approach:

$$P(BRY)=\frac{3}{9}*\frac{3}{8}*\frac{3}{7}*3!=\frac{9}{28}$$, we are multiplying by 3! since BRY scenario can occur in several different ways: BRY, RYB, RBY (# of permutations of 3 distinct letters BRY is 3!).

Combination approach:

$$P(BRY)=\frac{C^1_3*C^1_3*C^1_3}{C^3_9}=\frac{9}{28}$$.

Hope it's clear.

Hi Bunuel,

I have a question with the methodology here -- I can follow the work you're doing but i'm having a hard time understanding why. Can you please clarify the reasoning?

a) In your statement below, can you please elaborate WHY we multiply by 3!? -- [$$P=\frac{3}{9}*\frac{3}{8}*\frac{3}{7}*3!=\frac{9}{28}$$

If I reference the GMAT club book, I don't see it referencing the multiplication by a factorial(Page 109 of the book).

b) What is the difference between the method you used vs. doing the following? (9/9)(6/8)(3/7)

c) lastly, regarding your combinatorial approach, isn't that technically a reverse approach(pg 109 of the book)? Sorry if i'm being trivial here but i'm having a hard time connecting the dots hence the trivial questions. How can you use the 1C3 approach because don't you have 9 to begin with? Sorry - i'm going nuts here.
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Joined: 20 Feb 2016
Posts: 6
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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shahideh wrote:
We can choose 3 balls from 9 balls. Total possibilities: $$C^9_3$$
We should choose 1 ball from each color. Favorite possibilities: $$C^3_1*C^3_1*C^3_1$$

$$p=\frac{C^3_1*C^3_1*C^3_1}{C^9_3}=\frac{9}{28}$$
D

I'm trying to understand the basics, please excuse me if you find the question too silly:
Why don't we multiple by 3! in combinatorics method?
So first it's choosing i.e. 3C1 * 3C1* 3C1
Then why aren't the chosen balls arranged amongst themselves in 3! ways?
Math Expert V
Joined: 02 Aug 2009
Posts: 7765
Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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salonipatil wrote:
shahideh wrote:
We can choose 3 balls from 9 balls. Total possibilities: $$C^9_3$$
We should choose 1 ball from each color. Favorite possibilities: $$C^3_1*C^3_1*C^3_1$$

$$p=\frac{C^3_1*C^3_1*C^3_1}{C^9_3}=\frac{9}{28}$$
D

I'm trying to understand the basics, please excuse me if you find the question too silly:
Why don't we multiple by 3! in combinatorics method?
So first it's choosing i.e. 3C1 * 3C1* 3C1
Then why aren't the chosen balls arranged amongst themselves in 3! ways?

hi,
We are not doing so, because it will not effect the answer..
if you consider ORDER important, even the TOTAL ways will be multiplied by 3!..
ans = $$p=\frac{C^3_1*C^3_1*C^3_1*3!}{C^9_3*3!}=\frac{9}{28}$$
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Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3  [#permalink]

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_________________ Re: A basket contains 3 blue, 3 red and 3 yellow marbles. If 3   [#permalink] 10 Sep 2018, 13:47
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