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Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random are found to be 1 red and 1 black. If the probability that both balls come from bag A is 6/11 , then n is equal to ________.
- a)13
- b)6
- c)4
- d)3
Correct answer is option 'C'. Can you explain this answer?
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Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 b...
To solve this problem, we can use conditional probability. Let's break down the question step by step:
Step 1: Calculate the probability of choosing bag A or bag B
Since one bag is chosen at random, the probability of choosing bag A is P(A) = 1/2 and the probability of choosing bag B is P(B) = 1/2.
Step 2: Calculate the probability of drawing 1 red and 1 black ball from bag A
In bag A, there are 2 white, 1 black, and 3 red balls. The probability of drawing 1 red and 1 black ball from bag A can be calculated as follows:
P(red and black from A) = P(red from A) * P(black from A)
P(red from A) = 3/6 (since there are 3 red balls out of a total of 6 balls in bag A)
P(black from A) = 1/6 (since there is 1 black ball out of a total of 6 balls in bag A)
P(red and black from A) = (3/6) * (1/6) = 1/12
Step 3: Calculate the probability of drawing 1 red and 1 black ball from bag B
In bag B, there are n white, 3 black, and 2 red balls. The probability of drawing 1 red and 1 black ball from bag B can be calculated as follows:
P(red and black from B) = P(red from B) * P(black from B)
P(red from B) = 2/5 (since there are 2 red balls out of a total of 5 balls in bag B)
P(black from B) = 3/5 (since there are 3 black balls out of a total of 5 balls in bag B)
P(red and black from B) = (2/5) * (3/5) = 6/25
Step 4: Calculate the probability of drawing 1 red and 1 black ball regardless of the bag chosen
To calculate this probability, we need to consider both possibilities: drawing from bag A and drawing from bag B.
P(red and black) = P(A) * P(red and black from A) + P(B) * P(red and black from B)
P(red and black) = (1/2) * (1/12) + (1/2) * (6/25)
P(red and black) = 1/24 + 3/50
P(red and black) = 13/300
Step 5: Calculate the probability of choosing bag A given that 1 red and 1 black ball were drawn
Using conditional probability formula:
P(A|red and black) = P(A and red and black) / P(red and black)
We know that P(A and red and black) = 6/11 (given in the question).
P(A|red and black) = (6/11) / (13/300)
P(A|red and black) = (6/11) * (300/13)
P(A|red and black) = 1800/143
P(A|red and black) ≈ 12.59%
Step 6: Find the value of n
Since the probability of choosing bag A given that 1 red
Step 1: Calculate the probability of choosing bag A or bag B
Since one bag is chosen at random, the probability of choosing bag A is P(A) = 1/2 and the probability of choosing bag B is P(B) = 1/2.
Step 2: Calculate the probability of drawing 1 red and 1 black ball from bag A
In bag A, there are 2 white, 1 black, and 3 red balls. The probability of drawing 1 red and 1 black ball from bag A can be calculated as follows:
P(red and black from A) = P(red from A) * P(black from A)
P(red from A) = 3/6 (since there are 3 red balls out of a total of 6 balls in bag A)
P(black from A) = 1/6 (since there is 1 black ball out of a total of 6 balls in bag A)
P(red and black from A) = (3/6) * (1/6) = 1/12
Step 3: Calculate the probability of drawing 1 red and 1 black ball from bag B
In bag B, there are n white, 3 black, and 2 red balls. The probability of drawing 1 red and 1 black ball from bag B can be calculated as follows:
P(red and black from B) = P(red from B) * P(black from B)
P(red from B) = 2/5 (since there are 2 red balls out of a total of 5 balls in bag B)
P(black from B) = 3/5 (since there are 3 black balls out of a total of 5 balls in bag B)
P(red and black from B) = (2/5) * (3/5) = 6/25
Step 4: Calculate the probability of drawing 1 red and 1 black ball regardless of the bag chosen
To calculate this probability, we need to consider both possibilities: drawing from bag A and drawing from bag B.
P(red and black) = P(A) * P(red and black from A) + P(B) * P(red and black from B)
P(red and black) = (1/2) * (1/12) + (1/2) * (6/25)
P(red and black) = 1/24 + 3/50
P(red and black) = 13/300
Step 5: Calculate the probability of choosing bag A given that 1 red and 1 black ball were drawn
Using conditional probability formula:
P(A|red and black) = P(A and red and black) / P(red and black)
We know that P(A and red and black) = 6/11 (given in the question).
P(A|red and black) = (6/11) / (13/300)
P(A|red and black) = (6/11) * (300/13)
P(A|red and black) = 1800/143
P(A|red and black) ≈ 12.59%
Step 6: Find the value of n
Since the probability of choosing bag A given that 1 red
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Community Answer
Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 b...
P(1R and 1B) = P(A).P
⇒ n2 + 9n - 52 = 0
⇒ n = 4 is the only possible value.
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Bag A contains 2 white, 1 black and 3 red balls and bag B contains 3 black, 2 red and n white balls. One bag is chosen at random and 2 balls drawn from it at random are found to be 1 red and 1 black. If the probability that both balls come from bag A is 6/11 , then n is equal to ________.a)13b)6c)4d)3Correct answer is option 'C'. Can you explain this answer?
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