A coin is tossed. if it turns up head 2 balls will be drawn from urn A otherwise 2 balls will be drawn from a urn B. Urn A contains 3 black and 5 white balls. Urn B contains 7 black balls and 1 white ball. In both cases selections are to be made with replacement. what is the probability that Urn A is used given that both the balls drawn are black. The correct answer is 9/58?

Question

Answer

Problem: A coin is tossed. If it turns up head, 2 balls will be drawn from urn A otherwise 2 balls will be drawn from urn B. Urn A contains 3 black and 5 white balls. Urn B contains 7 black balls and 1 white ball. In both cases selections are to be made with replacement. What is the probability that Urn A is used given that both the balls drawn are black?

Solution:
Let's define the events:
- H: the coin turns up head
- T: the coin turns up tail
- A: 2 balls are drawn from urn A
- B: 2 balls are drawn from urn B
- BB: both balls drawn are black

We need to find P(A|BB), the probability that urn A is used given that both the balls drawn are black.

We can use Bayes' theorem to solve this problem:
P(A|BB) = P(BB|A) * P(A) / P(BB)

We know that P(A) = P(H) = 1/2, as the coin is fair. We also know that P(BB|A) is the probability of drawing 2 black balls from urn A, with replacement:
P(BB|A) = (3/8)^2 = 9/64

To calculate P(BB), we need to use the law of total probability:
P(BB) = P(BB|A) * P(A) + P(BB|B) * P(B)

We know that if the coin turns up tail, we draw 2 balls from urn B, so:
P(BB|B) = (7/8)^2 = 49/64
P(B) = P(T) = 1/2

Therefore:
P(BB) = (9/64) * (1/2) + (49/64) * (1/2) = 29/64

Now we can substitute all the values in Bayes' theorem:
P(A|BB) = (9/64) * (1/2) / (29/64) = 9/29

Therefore, the probability that urn A is used given that both the balls drawn are black is 9/29.

Answer

Similar Class 12 Doubts

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 bag contains 5 balls, three red and two white. Balls are randomly removed one at a time without replacement until all the red balls are drawn or all the white balls are drawn. The probability that the last ball drawn is white, is?

There are two containers P and Q. P contains 56 kg of salt and Q contains 36 kg of sugar. From P 24 kg of salt is taken out and poured into Q. Then 20kg of the mixture from Q is taken out and poured into P. Find the ratio of final quantity of salt to sugar in container P.

5 coins are tossed 128 times. What is the probability of getting 3 Or more heads and find out the expected Frequencies of 3 or more heads?

A coin is tossed 3time if heads occurs on first two toss find the probability of getting on third toss ?

Top Courses for Class 12

Suggested Free Tests

Related Content

Schools

Competitive Examinations

Quick Access Links

Technical Exams