Question

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is picked from B and put in A. Then a ball is drawn from A. Probability it is white is $p/q$. Find $p+q$.

• A. 23
• B. 22
• C. 21
• D. 24

Hint:

Use the Total Probability Theorem for multi-stage random experiments.

Answer 1

The Correct Option is A

To solve the problem, we need to determine the probability that a ball drawn from Bag A is white, after transferring one ball from Bag B to Bag A.

1. Initially, Bag A has 9 white and 8 black balls, and Bag B has 6 white and 4 black balls.
2. There are two possible cases when transferring a ball from Bag B to Bag A:
   • Case 1: A white ball is transferred from Bag B to Bag A.
   • Case 2: A black ball is transferred from Bag B to Bag A.
3. Calculate the probability for each case:
   • The probability of transferring a white ball from Bag B is \(\frac{6}{10}\), as there are 6 white balls out of a total of 10 balls in Bag B.
   • The probability of transferring a black ball from Bag B is \(\frac{4}{10}\), as there are 4 black balls out of a total of 10 balls in Bag B.
4. For each case, compute the probability that the drawn ball from Bag A is white:
   • Case 1 (White ball transferred):
     • Bag A will have 10 white balls and 8 black balls.
     • The probability of drawing a white ball from Bag A is \(\frac{10}{18}\). 
   • Case 2 (Black ball transferred):
     • Bag A will have 9 white balls and 9 black balls.
     • The probability of drawing a white ball from Bag A is \(\frac{9}{18}\).
5. Compute the total probability of drawing a white ball from Bag A using the law of total probability:
6. Simplify the expression:
7. The probability that a white ball is drawn from Bag A is \(\frac{8}{15}\). Here, \(p = 8\) and \(q = 15\), therefore \(p + q = 23\).

Thus, the value of \(p + q\) is 23. The correct answer is 23.