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# A bag contains 3 red, 4 black and 2 white balls. What is the

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A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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26 Jul 2004, 09:33
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A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?

(A) 2/27
(B) 1/9
(C) 1/3
(D) 4/27
(E) 2/9
[Reveal] Spoiler: OA

Last edited by Bunuel on 25 Oct 2017, 00:56, edited 2 times in total.
Renamed the topic, edited the question, added the OA and moved to PS forum.

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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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26 Jul 2004, 09:41
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A is correct if you want to get a Red ball First and then a White one.
If the order doesn't matter, then the answer is D.

(2/27)*2 = 4/27

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26 Jul 2004, 10:14
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This is how I came up with 2/27. the probability of getting one red is 3/9 (nine is the total number of balls). The probability of getting a white ball is 2/9 (9 again because the ball is put back after each draw) so 3/9*2/9 + 6/81 = 2/27

according to Dookie (who is right) if they are asking for the balls to draw one of the the other, which they are (successive draws) you have to multiply 2/27 by 2 = 4/27.

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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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31 Jul 2004, 06:53
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Here is how I solved it

First of all we have
Probability of drawing a Red ball is 3/9
Probability of drawing a White ball is 3/9

There are two ways in which the balls can be drawn

Case 1: Red ball in the first draw and white in the second draw
Hence the combined Probability is 3/9*2/9=6/81

Case 2: White ball in the first draw and red in the second draw
Hence the combined Probability is 2/9*3/9=6/81

both these cases satisfy our requirement
Hence either of them will do i.e OR
Hence the final probability comes to be
Case 1 OR Case 2 = 6/81 + 6/81 (OR means addition)
Hence the Ans is 12/81=4/27

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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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31 Jul 2004, 06:59
Sorry for the Typo above
the Probability of drawing the white ball is 2/9

I posted the above explanation but just forgot to login

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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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05 Aug 2004, 18:51
Dookie wrote:
A is correct if you want to get a Red ball First and then a White one.
If the order doesn't matter, then the answer is D.

(2/27)*2 = 4/27

Dookie-

First red, then white:
drawing red: 3/9
drawing white: 2/9

3/9 * 2/9 = 2/27.

First white, the red:
2/9 * 3/9 = 2/27

how come order being matter would change the result?

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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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05 Aug 2004, 18:58
D is the right answer, without any mention about the other, we need to assume that the order is independent so it is

3/9 * 2/9 * 2 ( last 2 is for the order ) = 4/27

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A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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23 Sep 2013, 06:53
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A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?

(A) 2/27
(B) 1/9
(C) 1/3
(D) 4/27
(E) 2/9

This is with replacement case.

$$P=2*\frac{3}{9}*\frac{2}{9}=\frac{4}{27}$$

We are multiplying by 2 as there are two possible wining scenarios RW and WR.

Answer: D.
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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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23 Sep 2013, 19:22
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Hello Bunuel,
I m a bit confused about when to consider and when not to consider. I am having a tough time understanding this concept. I was under the impression that in "with replacement" cases, we dont need to consider the cases, however in without replacement cases, scenarios needs to be considered.

In addition,I found this link, where the question is also testing the same concept, but we didn't consider the multiple cases here. Please clarify.
http://gmatclub.com/forum/rich-has-3-gr ... 55253.html

Can you provide a high level conceptual knowledge as in when to consider cases and when not to?
Pls help.

Posted from my mobile device
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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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27 Sep 2013, 19:21
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imhimanshu wrote:
Hello Bunuel,
I m a bit confused about when to consider and when not to consider. I am having a tough time understanding this concept. I was under the impression that in "with replacement" cases, we dont need to consider the cases, however in without replacement cases, scenarios needs to be considered.

In addition,I found this link, where the question is also testing the same concept, but we didn't consider the multiple cases here. Please clarify.
http://gmatclub.com/forum/rich-has-3-gr ... 55253.html

Can you provide a high level conceptual knowledge as in when to consider cases and when not to?
Pls help.

Posted from my mobile device

Responding to a pm:

The status of "replacement" has nothing to do with the "sequence". It only changes the probability.

Say we have 2 red and 3 white balls in a bag. We pull out two one after another with replacement. What is the probability that one is red and the other is white.
Now note that there are 4 ways in which you can pull out two balls from the bag:
1. You pull a Red and then a Red again RR - (2/5)*(2/5) (Note that it is with replacement)
2. You pull a Red and then a White RW - (2/5)*(3/5)
3. You pull a White and then a Red WR - (3/5)*(2/5)
4. You pull a White and then a White WW - (3/5)*(3/5)

Total probability = (2/5)*(2/5) + (2/5)*(3/5) + (3/5)*(2/5) + (3/5)*(3/5) = 1

In how many cases do we have a red and a white ball? In case 2 and case 3.
Probability of picking a red and a white with replacement = (2/5)*(3/5) + (3/5)*(2/5) = (3/5)*(2/5) * 2
Since the probability of picking a red and then a white is same as probability of picking a white and then a red, you simply write down one case and multiply it by 2. You do the same in case of 'without replacement' too. The only thing that changes is the probability.

Without Replacement:
1. You pull a Red and then a Red again RR - (2/5)*(1/4)
2. You pull a Red and then a White RW - (2/5)*(3/4)
3. You pull a White and then a Red WR - (3/5)*(2/4)
4. You pull a White and then a White WW - (3/5)*(2/4)
Probability of picking a red and a white WITHOUT replacement = (2/5)*(3/4) + (3/5)*(2/4) = (3/5)*(2/4) * 2

As for the link you have mentioned, this is exactly what is done there too. Check it out - I will show you how it is done there.
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Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Kudos [?]: 18480 [2], given: 237 Current Student Joined: 20 Jan 2014 Posts: 175 Kudos [?]: 78 [0], given: 120 Location: India Concentration: Technology, Marketing Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink] ### Show Tags 22 Sep 2014, 05:33 VeritasPrepKarishma wrote: imhimanshu wrote: Hello Bunuel, I m a bit confused about when to consider and when not to consider. I am having a tough time understanding this concept. I was under the impression that in "with replacement" cases, we dont need to consider the cases, however in without replacement cases, scenarios needs to be considered. In addition,I found this link, where the question is also testing the same concept, but we didn't consider the multiple cases here. Please clarify. http://gmatclub.com/forum/rich-has-3-gr ... 55253.html Can you provide a high level conceptual knowledge as in when to consider cases and when not to? Pls help. Posted from my mobile device Responding to a pm: The status of "replacement" has nothing to do with the "sequence". It only changes the probability. Say we have 2 red and 3 white balls in a bag. We pull out two one after another with replacement. What is the probability that one is red and the other is white. Now note that there are 4 ways in which you can pull out two balls from the bag: 1. You pull a Red and then a Red again RR - (2/5)*(2/5) (Note that it is with replacement) 2. You pull a Red and then a White RW - (2/5)*(3/5) 3. You pull a White and then a Red WR - (3/5)*(2/5) 4. You pull a White and then a White WW - (3/5)*(3/5) Total probability = (2/5)*(2/5) + (2/5)*(3/5) + (3/5)*(2/5) + (3/5)*(3/5) = 1 In how many cases do we have a red and a white ball? In case 2 and case 3. Probability of picking a red and a white with replacement = (2/5)*(3/5) + (3/5)*(2/5) = (3/5)*(2/5) * 2 Since the probability of picking a red and then a white is same as probability of picking a white and then a red, you simply write down one case and multiply it by 2. You do the same in case of 'without replacement' too. The only thing that changes is the probability. Without Replacement: 1. You pull a Red and then a Red again RR - (2/5)*(1/4) 2. You pull a Red and then a White RW - (2/5)*(3/4) 3. You pull a White and then a Red WR - (3/5)*(2/4) 4. You pull a White and then a White WW - (3/5)*(2/4) Probability of picking a red and a white WITHOUT replacement = (2/5)*(3/4) + (3/5)*(2/4) = (3/5)*(2/4) * 2 As for the link you have mentioned, this is exactly what is done there too. Check it out - I will show you how it is done there. Thank You Karishma. I got a key concept here But i am more comfortable by Combination method 1C3*1C2/ (1C9 * 1C9) = 6/81 = 2/27 Now we can get this in two ways (as described by u) 2* 2/27 = 4/27 _________________ Consider +1 Kudos Please Kudos [?]: 78 [0], given: 120 Intern Joined: 28 Dec 2015 Posts: 42 Kudos [?]: 3 [0], given: 62 Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink] ### Show Tags 20 Jun 2016, 00:10 P(RW)---Remember that successive draws means one after another and it is not necessary that red will come before white,it can be vice versa too. So,First draw Red=3/9 Second draw White=2/9, Total number of balls will not change since we are putting back the balls in the bag. P(R and W)=3/9*2/9*2!=4/27 2!=RW or WR,any way you can draw the ball Kudos [?]: 3 [0], given: 62 Senior Manager Joined: 02 Mar 2012 Posts: 356 Kudos [?]: 94 [0], given: 4 Schools: Schulich '16 Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink] ### Show Tags 20 Jun 2016, 05:31 answer will be D. If the question included the word'RESPECTIVELY' order would matter ..Then answer would be A if nothing is given order does not matter hope it helps Kudos [?]: 94 [0], given: 4 SVP Joined: 06 Nov 2014 Posts: 1904 Kudos [?]: 564 [0], given: 23 Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink] ### Show Tags 16 Jul 2016, 11:45 Keen wrote: A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn? (A) 2/27 (B) 1/9 (C) 1/3 (D) 4/27 (E) 2/9 Please explain your answer. I came up with [Reveal] Spoiler: A but that is wrong. 3R, 4B, 2W balls. P(1R, 1W) = (3/9)*(2/9) = 1/3*2/9 = 2/27 Now the Red and the While balls can be drawn in any order Hence probability = 2*2/27 = 4/27 Correct Option: D Kudos [?]: 564 [0], given: 23 Intern Joined: 14 Aug 2017 Posts: 29 Kudos [?]: 1 [0], given: 12 Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink] ### Show Tags 25 Oct 2017, 00:52 karishma Bunuel - Can you please explain as to where my concept is headed in the wrong direction? At first i saw that there are 9 balls and 2 balls are to be drawn, i selected the 2 balls out of 9 (9C2) giving me the total outcomes and next when i am supposed to draw the balls in the manner of white and red it clicked to me that there are 4 case, 1 - RR 2 - RW 3 - WR 4 - WW and i applied the same logic to the above 4 cases, but the answer did not match? Can you please help me with this. Thanks in advance. Kudos [?]: 1 [0], given: 12 Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7867 Kudos [?]: 18480 [1], given: 237 Location: Pune, India Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink] ### Show Tags 26 Oct 2017, 21:58 1 This post received KUDOS Expert's post siddyj94 wrote: karishma Bunuel - Can you please explain as to where my concept is headed in the wrong direction? At first i saw that there are 9 balls and 2 balls are to be drawn, i selected the 2 balls out of 9 (9C2) giving me the total outcomes and next when i am supposed to draw the balls in the manner of white and red it clicked to me that there are 4 case, 1 - RR 2 - RW 3 - WR 4 - WW and i applied the same logic to the above 4 cases, but the answer did not match? Can you please help me with this. Thanks in advance. Note that you cannot use 9C2 here because you are drawing balls with replacement. 9C2 means draw 2 balls out of 9 or draw 1 out of 9 and then 1 out of 8. But here, we need to draw 1 out of 9 and then again 1 out of 9 (since the first ball is put back) Also, why only 4? Wouldn't we have other cases too such as a Red and a Black? Look at the solutions above to see how to solve it. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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29 Oct 2017, 06:41
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Keen wrote:
A bag contains 3 red, 4 black and 2 white balls. What is the probability of drawing a red and a white ball in two successive draws, each ball being put back after it is drawn?

(A) 2/27
(B) 1/9
(C) 1/3
(D) 4/27
(E) 2/9

There are 9 balls in the bag, so the probability of drawing a red ball is P(Red) = 3/9 = ⅓, and the probability of drawing a white ball is P(White) = 2/9. We will draw two balls, replacing each ball after it is drawn.

The probability of drawing a red ball first and then a white ball is: P(Red) x P(White) = ⅓ x 2/9 = 2/27. But we can also draw a white ball first and then a red ball: P(White) x P(Red) = 2/9 x ⅓ = 2/27. Either of these outcomes satisfies our outcome of interest, and so we add the two probabilities: 2/27 + 2/27 = 4/27.

Answer: D
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Re: A bag contains 3 red, 4 black and 2 white balls. What is the [#permalink]

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25 Nov 2017, 14:15
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Hi All,

We're told that a bag contains 3 red, 4 black and 2 white balls. We're asked for the probability of drawing a red and a white ball in two successive draws, while replacing each ball after it is drawn.

The prompt does NOT state that the red ball has to be drawn first, so there are two options that we have to consider: red first, white second and white first, red second.

The probability of pulling a red first and a white second (with replacement) = (3/9)(2/9) = 6/81
The probability of pulling a while first and a red second (with replacement) = (2/9)(3/9) = 6/81
The total probability of pulling a red and white ball is 6/81 + 6/81 = 12/81 = 4/27

Final Answer:
[Reveal] Spoiler:
D

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