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Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh

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Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh  [#permalink]

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New post 12 Nov 2019, 03:21
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Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls. A white ball is picked from bag A and put into bag B. Then, three balls are picked from bag B and put into bag A. Find the probability that a ball picked now from bag A is black.

A. 1/4
B. 1/3
C. 7/12
D. 5/12
E. 11/24


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Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh  [#permalink]

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New post Updated on: 12 Nov 2019, 13:31
Bag A = 6 white and 4 black = 10 total
Bag B = 3 white and 2 black = 5 total
now
Bag A = 5 white and 4 black = 9 total
Bag B = 4 white and 2 black = 6 total

three balls are picked from bag B and put into bag A Find the probability that a ball picked now from bag A is black.

case 1 ; all three white = Bag A ; 8 white + 4 black ; 4/12 * 4c3/6c3 = 1/15
case 2 ; 2 white and 1 black ; Bag A ; 7 white + 5 black ; 5/12* 4c2*2c1/6c3= 1/4
case 3 ; 1 white and 2 black ; Bag A; 6 white + 6 black ; 6/12 *4c1*2c2/6c3= 1/10
total probability that a ball picked now from bag A is black = 1/15+1/4+1/10= 5/12
IMO D


Bunuel wrote:
Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls. A white ball is picked from bag A and put into bag B. Then, three balls are picked from bag B and put into bag A. Find the probability that a ball picked now from bag A is black.

A. 1/4
B. 1/3
C. 7/12
D. 5/12
E. 11/24


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Originally posted by Archit3110 on 12 Nov 2019, 03:47.
Last edited by Archit3110 on 12 Nov 2019, 13:31, edited 1 time in total.
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Re: Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh  [#permalink]

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New post 12 Nov 2019, 12:58
archit Always remember: probability can never be greater than 1 (at least on Earth ;)

It's a conditional probability question.

Number of balls we have after First transfer

Bag A = 5 white and 4 black
Bag B = 4 white and 2 black

Now there are 3 cases possible for 2nd transfer (as you mentioned)

1. when all 3 picks from bag B are white, then probability of a ball picked now from bag A is black.

\(\frac{4C3}{6C3} * \frac{4}{12}= \frac{1}{15}\)

2. when 2 picks from bag B are white and 1 pick is black, then probability of a ball picked now from bag A is black.

\(\frac{4C2*2C1}{6C3} * \frac{5}{12}= \frac{1}{4}\)

3. when 1 pick from bag B is white and 2 picks are black, then probability of a ball picked now from bag A is black.

\(\frac{4C1*2C2}{6C3} * \frac{6}{12}= \frac{1}{10}\)

Total probability= \(\frac{1}{15}+\frac{1}{4}+\frac{1}{10}= \frac{4+15+6}{60}= \frac{5}{12}\)









Archit3110 wrote:
Bag A = 6 white and 4 black = 10 total
Bag B = 3 white and 2 black = 5 total
now
Bag A = 5 white and 4 black = 9 total
Bag B = 4 white and 2 black = 6 total

three balls are picked from bag B and put into bag A Find the probability that a ball picked now from bag A is black.

case 1 ; all three white = Bag A ; 8 white + 4 black ; 4/12
case 2 ; 2 white and 1 black ; Bag A ; 7 white + 5 black ; 5/12
case 3 ; 1 white and 2 black ; Bag A; 6 white + 6 black ; 6/12
total probability that a ball picked now from bag A is black = 4/12 +5/12+6/12 ; 15/12 ; 5/4


Bunuel wrote:
Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls. A white ball is picked from bag A and put into bag B. Then, three balls are picked from bag B and put into bag A. Find the probability that a ball picked now from bag A is black.

A. 1/4
B. 1/3
C. 7/12
D. 5/12
E. 11/24


Are You Up For the Challenge: 700 Level Questions
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Re: Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh  [#permalink]

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New post 12 Nov 2019, 13:07
nick1816
I posted the solution incomplete the combination part along each condition got left out my bad.. ?

nick1816 wrote:
archit Always remember: probability can never be greater than 1 (at least on Earth ;)

It's a conditional probability question.

Number of balls we have after First transfer

Bag A = 5 white and 4 black
Bag B = 4 white and 2 black

Now there are 3 cases possible for 2nd transfer (as you mentioned)

1. when all 3 picks from bag B are white, then probability of a ball picked now from bag A is black.

\(\frac{4C3}{6C3} * \frac{4}{12}= \frac{1}{15}\)

2. when 2 picks from bag B are white and 1 pick is black, then probability of a ball picked now from bag A is black.

\(\frac{4C2*2C1}{6C3} * \frac{5}{12}= \frac{1}{4}\)

3. when 1 pick from bag B is white and 2 picks are black, then probability of a ball picked now from bag A is black.

\(\frac{4C1*2C2}{6C3} * \frac{6}{12}= \frac{1}{10}\)

Total probability= \(\frac{1}{15}+\frac{1}{4}+\frac{1}{10}= \frac{4+15+6}{60}= \frac{5}{12}\)









Archit3110 wrote:
Bag A = 6 white and 4 black = 10 total
Bag B = 3 white and 2 black = 5 total
now
Bag A = 5 white and 4 black = 9 total
Bag B = 4 white and 2 black = 6 total

three balls are picked from bag B and put into bag A Find the probability that a ball picked now from bag A is black.

case 1 ; all three white = Bag A ; 8 white + 4 black ; 4/12
case 2 ; 2 white and 1 black ; Bag A ; 7 white + 5 black ; 5/12
case 3 ; 1 white and 2 black ; Bag A; 6 white + 6 black ; 6/12
total probability that a ball picked now from bag A is black = 4/12 +5/12+6/12 ; 15/12 ; 5/4


Bunuel wrote:
Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls. A white ball is picked from bag A and put into bag B. Then, three balls are picked from bag B and put into bag A. Find the probability that a ball picked now from bag A is black.

A. 1/4
B. 1/3
C. 7/12
D. 5/12
E. 11/24


Are You Up For the Challenge: 700 Level Questions


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Re: Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh  [#permalink]

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New post 13 Nov 2019, 05:25

Solution



Given
    • Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls.
    • A white ball is picked from bag A and put into bag B.
      o Then, three balls are picked from bag B and put into bag A.

To find
    • The probability that a ball picked now from bag A is black.

Approach and Working out
    • Bag A contains 6 white balls and 4 black balls and bag B contains 3 white balls and 2 black balls
    • A white ball is picked from bag A and put into bag B
      o So, Bag A - 5 white balls and 4 black balls
      o Bag B - 4 white and 2 black balls.
    • Then, three balls are picked from bag B and put into bag A
      o There will be 3 cases in which the balls from bag B can be picked.
         2 black and 1 white
          • Bag A – 6 White and 6 black
          • Probability = \(\frac{4c1 * 2c2 }{6c3} * \frac{6}{12}\) = \(\frac{1}{10}\)
         1 black and 2 white
          • Bag A – 7 White and 5 black
          • Probability = \(\frac{4c2 * 2c1 }{6c3} * \frac{6}{12}\) = \(\frac{1}{4}\)
         3 white
          • Bag A – 8 White and 4 black
          • Probability = \(\frac{4c3 }{6c3} * \frac{6}{12}\) = \(\frac{1}{15}\)
    • Total probability= 1/10 +1 /4 + 1/15 =5/12


Thus, option D is the correct answer.
Correct Answer: Option D
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Re: Bag A contains 6 white balls and 4 black balls and bag B contains 3 wh   [#permalink] 13 Nov 2019, 05:25
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