Question

Bag \( A \) contains 9 white and 8 black balls and bag \( B \) contains 6 white and 4 black balls. A ball is randomly transferred from bag \( B \) to bag \( A \), then a ball is drawn from bag \( A \). If the probability that the drawn ball is white is \( \dfrac{p}{q} \) (where \( p \) and \( q \) are coprime), then find \( p + q \):

Hint:

In transfer problems, always split into cases based on what is transferred and then apply total probability.

Answer 1

Correct Answer: 23

To solve this problem, we need to find the probability that a drawn ball from bag \( A \) is white after transferring a ball from bag \( B \) to bag \( A \). We proceed with the following steps: 
 

1. Initial setup:
   • Bag \( A \) initially has 9 white and 8 black balls.
   • Bag \( B \) has 6 white and 4 black balls.
2. Possible scenarios:
   A ball is selected from bag \( B \) and transferred to bag \( A \). There are two scenarios:
   • A white ball is transferred from \( B \).
   • A black ball is transferred from \( B \).
3. Calculate probabilities:
   First, calculate the probability of transferring each type of ball:
   • Probability of transferring a white ball from \( B \): \(\dfrac{6}{10}\).
   • Probability of transferring a black ball from \( B \): \(\dfrac{4}{10}\).
4. Calculate configurations after transfer:
   Analyze the composition of bag \( A \) after each possible transfer:
   • If a white ball is transferred: Bag \( A \) will have 10 white and 8 black balls.
   • If a black ball is transferred: Bag \( A \) will have 9 white and 9 black balls.
5. Calculate final probabilities of drawing a white ball:
   • Drawing a white ball if a white ball was transferred: \(\dfrac{10}{18}\).
   • Drawing a white ball if a black ball was transferred: \(\dfrac{9}{18}\).
6. Combine probabilities using the law of total probability:
   Using the probabilities from above:
    

\(\text{Probability (white from A)} = \left(\dfrac{6}{10} \cdot \dfrac{10}{18}\right) + \left(\dfrac{4}{10} \cdot \dfrac{9}{18}\right)\)

1. Simplify the expression:
   • \(\dfrac{6 \cdot 10}{10 \cdot 18} + \dfrac{4 \cdot 9}{10 \cdot 18} = \dfrac{60}{180} + \dfrac{36}{180} = \dfrac{96}{180} = \dfrac{16}{30}\).
2. Sum the coprime numbers:
   If the probability is \(\dfrac{8}{15}\), we have \(p = 8\) and \(q = 15\). Thus, \(p + q = 23\).

The calculated value of \(p + q\) is 23, which falls within the provided range of 23 to 23, confirming the solution is correct.