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

Step 1: Consider the transfer from bag \( B \).
From bag \( B \), \[ P(\text{White transferred}) = \frac{6}{10}, \quad P(\text{Black transferred}) = \frac{4}{10} \]
Step 2: Case I – White ball transferred.
Bag \( A \) now has 10 white and 8 black balls. \[ P(\text{White drawn} \mid \text{White transferred}) = \frac{10}{18} \]
Step 3: Case II – Black ball transferred.
Bag \( A \) now has 9 white and 9 black balls. \[ P(\text{White drawn} \mid \text{Black transferred}) = \frac{9}{18} \]
Step 4: Apply the law of total probability.
\[ P(\text{White drawn}) = \frac{6}{10}\cdot\frac{10}{18} + \frac{4}{10}\cdot\frac{9}{18} \] \[ = \frac{60 + 36}{180} = \frac{96}{180} = \frac{8}{15} \]
Step 5: Final Answer.
Here \( p = 8 \), \( q = 15 \). \[ p + q = 23 \]