Amy has two bags. Bag I has 7 red and 4 blue balls and bag II has 5 red and 9 blueballs. Amy draws a ball at random and it turns out to be red.Determine the probability that the ball was from the bag I using the Bayes
Question
Amy has two bags. Bag I has 7 red and 4 blue balls and bag II has 5 red and 9 blueballs. Amy draws a ball at random and it turns out to be red.Determine the probability that the ball was from the bag I using the Bayes
Solution
To solve this problem, we need to use Bayes' theorem, which is a way to find a probability when we know certain other probabilities. The formula is:
P(A|B) = P(B|A) * P(A) / P(B)
Where:
- P(A|B) is the probability we are trying to find: the ball is from Bag I given that it is red.
- P(B|A) is the probability that the ball is red given that it is from Bag I.
- P(A) is the probability that the ball is from Bag I.
- P(B) is the probability that the ball is red.
Step 1: Calculate P(B|A) There are 7 red balls in Bag I out of a total of 11 balls. So, P(B|A) = 7/11.
Step 2: Calculate P(A) There are 2 bags, so the probability that the ball is from Bag I is 1/2. So, P(A) = 1/2.
Step 3: Calculate P(B) The total number of red balls is 7 (from Bag I) + 5 (from Bag II) = 12. The total number of balls is 11 (from Bag I) + 14 (from Bag II) = 25. So, P(B) = 12/25.
Step 4: Substitute these values into Bayes' theorem and solve: P(A|B) = (7/11 * 1/2) / (12/25) = 0.5833 or 58.33%.
So, the probability that the ball was from Bag I given that it is red is approximately 58.33%.
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