Bag 1 contains 3 red and 5 black balls while another Bag 2 contains 4 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it is drawn from bag 2.a)31/62b)16/62c)16/31d)31/32Correct answer is option 'C'. Can you explain this answer?

Question

Answer

Problem:
Bag 1 contains 3 red and 5 black balls while another Bag 2 contains 4 red and 6 black balls. One ball is drawn at random from one of the bags and it is found to be red. Find the probability that it is drawn from bag 2.

Solution:
Let A be the event that the ball is drawn from bag 1, and B be the event that the ball is drawn from bag 2. We want to find the probability of B given that a red ball has been drawn, that is, P(B|red).

By Bayes' theorem, we have:

P(B|red) = P(red|B) P(B) / P(red)

We know that P(red|B) is the probability of drawing a red ball from bag 2, which is 4/10 = 2/5. P(B) is the probability of drawing from bag 2, which is 1/2 since each bag is equally likely to be chosen.

To find P(red), we use the law of total probability:

P(red) = P(red|A) P(A) + P(red|B) P(B)

P(red|A) is the probability of drawing a red ball from bag 1, which is 3/8. P(A) is the probability of drawing from bag 1, which is also 1/2. We already know that P(red|B) is 2/5 and P(B) is 1/2. Substituting these values, we get:

P(red) = (3/8)(1/2) + (2/5)(1/2) = 31/80

Now we can substitute all the values in the Bayes' theorem formula:

P(B|red) = (2/5)(1/2) / (31/80) = 16/31

Therefore, the probability that the ball is drawn from bag 2 given that it is red is 16/31.

Answer

Let E1 = event of choosing the bag 1, E2 = event of choosing the bag 2.
Let A be event of drawing a red ball.
P(E1) = P(E2) = 1/2.
Also, P(A | E1) = P(drawing a red ball from Bag 1) = 3/8.
And P(A | E2) = P(drawing a red ball from Bag 2) = 4/10.
The probability of drawing a ball from bag 2, being given that it is red is P(E2 | A).
By using Bayes’ theorem,
P(E2 | A) = P(E2)P(A | E2)/( P(E1)P(A│E1)+P(E2)P(A | E2))
= (1/2 × 4/10) / ((1/2 × 3/8)) + (1/2 × 4/10)) = 16/31.

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