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Manager  Joined: 28 Jul 2004
Posts: 123
Location: Melbourne
Schools: Yale SOM, Tuck, Ross, IESE, HEC, Johnson, Booth
A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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10
36 00:00

Difficulty:   55% (hard)

Question Stats: 61% (01:33) correct 39% (02:31) wrong based on 455 sessions

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A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

_________________
Math Expert V
Joined: 02 Sep 2009
Posts: 58100
Re: What is the Probability of Red Ball?  [#permalink]

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10
14
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

Thanks

The initial probability of drawing red ball is 2/8. Without knowing the other results, the probability of drawing red ball will not change for ANY successive drawing: second, third, fourth...

Similar problems:
probability-of-picking-last-97015.html?hilit=initial%20probability

Hope it's clear.
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Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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4
2
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

Responding to a pm:

The point is that the probability of picking a red will not depend on which draw it is.

In the first draw, probability of picking a red is 2/8 = 1/4

Probability of picking a red in the second draw:

2 cases:

Case 1: First draw is non red.
Probability = (6/8)*(2/7)
This is equal to (2/8)(6/7). Think about it: Probability of picking non red first and then red will be the same as probability of picking a red first and then a non red.

Case 2: First draw is red.
Probability = (2/8)*(1/7)
This is the probability of picking a red first and then a red again.

Total probability of second draw being red = (2/8)*(6/7) + (2/8)*(1/7) = 2/8
This is just the probability of picking a red first and then any ball (non red or red). Probability of picking ANY ball will be 1. Hence, the probability of picking a red in the second draw will be the same as the probability of picking a red in the first draw.

Similarly probability of picking a red in any draw will be the same.
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Karishma
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Re: What is the Probability of Red Ball?  [#permalink]

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6
In problems such as these, the answer is just the probability based on the initial population... so 2/8 and it would be the same answer regardless of the nth try, 1st, 5th, 8th...
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Manager  Joined: 25 Jun 2010
Posts: 82
Re: What is the Probability of Red Ball?  [#permalink]

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2
2
Favorable outcomes :

WWR -> P(WWR)= 3/8 * 2/7 * 2/6 = 12/336
WRR -> P(WRR)= 3/8 * 2/7 * 1/6 = 6/336
RWR -> P(RWR)= 2/8 * 3/7 * 1/6 = 6/336
BBR -> P(BBR)= 3/8 * 2/7 * 2/6 = 12/336
BRR -> P(BRR)= 3/8 * 2/7 * 1/7 = 6/336
RBR -> P(RBR)= 2/8 * 3/7 * 1/6 = 6/336
WBR -> P(WBR)= 3/8 * 3/7 * 2/6 = 18/336
BWR -> P(BWR)= 3/8 * 3/7 * 2/6 = 18/336

So, P(Total) = 84/336 = 1/4 = 0.25
Manager  Joined: 25 Jun 2010
Posts: 82
Re: What is the Probability of Red Ball?  [#permalink]

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Excellent Bunuel!
I have seen a couple of such nice, handful and timesavers scattered across different posts . Have you considered posting a sticky where you collect them at one place.
That would be very helpful , also the GMatClub may add it to the math section of the IPhone app.

Thanks Posted from GMAT ToolKit
Manager  Joined: 16 Feb 2010
Posts: 166
Re: What is the Probability of Red Ball?  [#permalink]

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2
Bunuel wrote:
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

Thanks

The initial probability of drawing red ball is 2/8. Without knowing the other results, the probability of drawing red ball will not change for ANY successive drawing: second, third, fourth...

Similar problems:
probability-of-picking-last-97015.html?hilit=initial%20probability

Hope it's clear.

!!!!!
went the long way - still under 2min

RedOtherRed +OtherRR+OOR

$$\frac{1}{28}+\frac{1}{28}+\frac{5}{28}=\frac{7}{28}=\frac{1}{4}$$
Manager  Joined: 14 Jun 2010
Posts: 169
Re: What is the Probability of Red Ball?  [#permalink]

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2
P(drawing a red ball) = 1-P(not drawing a red ball )
=> 1-6C1/8C1 = 1/4
Manager  Joined: 16 Feb 2010
Posts: 166
Re: What is the Probability of Red Ball?  [#permalink]

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psychomath wrote:
P(drawing a red ball) = 1-P(not drawing a red ball )
=> 1-6C1/8C1 = 1/4

IMO when someone masters the ability to use combinations and probability at the same time in a gmat question, he/she has reached the level to move to a different topic....

@psychomath nice solution...
Senior Manager  Joined: 07 Sep 2010
Posts: 251
Re: A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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Hi Karishma,

Thank You for the explanation.

Just curious to know, do we have any scenario where this phenomenon fails.

Thanks

VeritasPrepKarishma wrote:

The point is that the probability of picking a red will not depend on which draw it is.

Manager  Joined: 04 Apr 2013
Posts: 114
Re: What is the Probability of Red Ball?  [#permalink]

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Bunuel wrote:
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

Thanks

The initial probability of drawing red ball is 2/8. Without knowing the other results, the probability of drawing red ball will not change for ANY successive drawing: second, third, fourth...

Similar problems:
probability-of-picking-last-97015.html?hilit=initial%20probability

Hope it's clear.

Bunuel,

Why is it not
7c2 * 1 / 8c3 = 3/8?

selecting any 2 without 1 red is 7c2....selecting the last red is 1...so fav outcomes = 7c2 * 1

total outcomes = 8c3

prob = 7c2/8c3

Where am I going wrong?

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MGMAT1 - 540 ( Trying to improve )
Math Expert V
Joined: 02 Sep 2009
Posts: 58100
Re: What is the Probability of Red Ball?  [#permalink]

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Bunuel wrote:
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

Thanks

The initial probability of drawing red ball is 2/8. Without knowing the other results, the probability of drawing red ball will not change for ANY successive drawing: second, third, fourth...

Similar problems:
probability-of-picking-last-97015.html?hilit=initial%20probability

Hope it's clear.

Bunuel,

Why is it not
7c2 * 1 / 8c3 = 3/8?

selecting any 2 without 1 red is 7c2....selecting the last red is 1...so fav outcomes = 7c2 * 1

total outcomes = 8c3

prob = 7c2/8c3

Where am I going wrong?

Don't quite understand what you are doing there. What does "selecting any 2 without 1 red is 7c2" mean?

Anyway, the question basically asks: if we put all 8 balls in a row, what is the probability that the third ball is red?

Similar questions might help:
a-box-contains-3-yellow-balls-and-5-black-balls-one-by-one-90272.html
a-bag-contains-3-white-balls-3-black-balls-2-red-balls-100023.html
each-of-four-different-locks-has-a-matching-key-the-keys-101553.html
if-40-people-get-the-chance-to-pick-a-card-from-a-canister-97015.html
new-set-of-mixed-questions-150204-100.html#p1208473
a-medical-researcher-must-choose-one-of-14-patients-to-127396.html
_________________
Manager  Joined: 04 Apr 2013
Posts: 114
Re: What is the Probability of Red Ball?  [#permalink]

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Bunuel,

first 2 balls can be selected from any of [3 black + 3 white + 1 red (7 balls)] - so selecting any 2 balls from 7 balls is 7c2.

probability of selecting the remaining red ball is 1.

so fav outcomes = 7c2*1.

probability = 7c2/8c3 = 3/8/

I know my thinking is definitely wrong.

Can you please point out the defect in my thinking?
_________________

MGMAT1 - 540 ( Trying to improve )
Veritas Prep GMAT Instructor D
Joined: 16 Oct 2010
Posts: 9637
Location: Pune, India
Re: A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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2
1
VeritasPrepKarishma wrote:
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

Responding to a pm:

The point is that the probability of picking a red will not depend on which draw it is.

In the first draw, probability of picking a red is 2/8 = 1/4

Probability of picking a red in the second draw:

2 cases:

Case 1: First draw is non red.
Probability = (6/8)*(2/7)
This is equal to (2/8)(6/7). Think about it: Probability of picking non red first and then red will be the same as probability of picking a red first and then a non red.

Case 2: First draw is red.
Probability = (2/8)*(1/7)
This is the probability of picking a red first and then a red again.

Total probability of second draw being red = (2/8)*(6/7) + (2/8)*(1/7) = 2/8
This is just the probability of picking a red first and then any ball (non red or red). Probability of picking ANY ball will be 1. Hence, the probability of picking a red in the second draw will be the same as the probability of picking a red in the first draw.

Similarly probability of picking a red in any draw will be the same.

Responding to a pm:

Quote:
Please tell me what is wrong with my thinking.

first 2 balls can be selected from any of [3 black + 3 white + 1 red (7 balls)] - so selecting any 2 balls from 7 balls is 7c2.

probability of selecting the remaining red ball is 1. (You are mixing combinations with probability.)

so fav outcomes = 7c2*1.

probability = 7c2/8c3 = 3/8

I know my thinking is definitely wrong.

Can you please point out the defect in my thinking?

There are a lot of problems here.

If you want to use combinations here, you can do this:

Assuming all balls are distinct, Number of ways of selecting 3 balls one after another without replacement = 7*6*1*2
(Keep one red ball away for the third pick. This can be done in 2 ways. Now of the 7 remaining balls, pick 1 for the first pick and then another for the second pick.)

Total number of outcomes = 8*7*6

Probability = (7*6*2)/(8*7*6) = 1/4
_________________
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Intern  Joined: 27 Jul 2013
Posts: 4
Re: What is the Probability of Red Ball?  [#permalink]

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Bunuel,

first 2 balls can be selected from any of [3 black + 3 white + 1 red (7 balls)] - so selecting any 2 balls from 7 balls is 7c2.

probability of selecting the remaining red ball is 1.

so fav outcomes = 7c2*1.

probability = 7c2/8c3 = 3/8/

I know my thinking is definitely wrong.

Can you please point out the defect in my thinking?

The big mistake is here that you ignored the phrase of "drawing without replacement", then your probability didn't exclude 1 ball picked up previously. Intern  Joined: 27 Jul 2013
Posts: 4
Re: What is the Probability of Red Ball?  [#permalink]

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Bunuel,

first 2 balls can be selected from any of [3 black + 3 white + 1 red (7 balls)] - so selecting any 2 balls from 7 balls is 7c2.

probability of selecting the remaining red ball is 1.

so fav outcomes = 7c2*1.

probability = 7c2/8c3 = 3/8/

I know my thinking is definitely wrong.

Can you please point out the defect in my thinking?

The big mistake is here that you ignored the phrase of "drawing without replacement", then your probability didn't exclude 1 ball picked up previously. Senior Manager  Joined: 10 Jul 2013
Posts: 294
Re: A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

the probability of the 3rd ball not to be Red = 6/8
so, the required probability = 1- 6/8 = 2/8 = 0.25
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SVP  Joined: 06 Sep 2013
Posts: 1604
Concentration: Finance
Re: A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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Asifpirlo wrote:
krishan wrote:
A bag contains 3 white balls, 3 black balls & 2 red balls. One by one three balls are drawn out without replacement. What is the probability that the third ball is red?

A. 0.25
B. 0.15
C. 0.35
D. 0.45
E. 0.40

the probability of the 3rd ball not to be Red = 6/8
so, the required probability = 1- 6/8 = 2/8 = 0.25

Not quite sure because after 2 draws there won't be 8 balls anymore no?
Also, what happens if we draw a red in the first two? Then won't the probability of drawing a red in the third draw be zero?

Cheers
J
Intern  Joined: 01 Jan 2014
Posts: 5
Location: United States
A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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mainhoon wrote:
In problems such as these, the answer is just the probability based on the initial population... so 2/8 and it would be the same answer regardless of the nth try, 1st, 5th, 8th...

By "In problems such as these", you meant "Replacement Problems", right?
Just want to be sure so that I can apply it for similar problems. : )
Math Expert V
Joined: 02 Sep 2009
Posts: 58100
Re: A bag contains 3 white balls, 3 black balls & 2 red balls  [#permalink]

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Sen1120 wrote:
mainhoon wrote:
In problems such as these, the answer is just the probability based on the initial population... so 2/8 and it would be the same answer regardless of the nth try, 1st, 5th, 8th...

By "In problems such as these", you meant "Replacement Problems", right?
Just want to be sure so that I can apply it for similar problems. : )

Problems such as these:
a-box-contains-3-yellow-balls-and-5-black-balls-one-by-one-90272.html
a-bag-contains-3-white-balls-3-black-balls-2-red-balls-100023.html
each-of-four-different-locks-has-a-matching-key-the-keys-101553.html
if-40-people-get-the-chance-to-pick-a-card-from-a-canister-97015.html
new-set-of-mixed-questions-150204-100.html#p1208473
a-medical-researcher-must-choose-one-of-14-patients-to-127396.html
_________________ Re: A bag contains 3 white balls, 3 black balls & 2 red balls   [#permalink] 02 Sep 2014, 20:13

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