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

Probability of transferring White ($T_W$) from B = $6/10 = 3/5$.
Probability of transferring Black ($T_B$) from B = $4/10 = 2/5$.
If $T_W$ occurs, Bag A has 10W, 8B (18 total). P(W|A) = $10/18 = 5/9$.
If $T_B$ occurs, Bag A has 9W, 9B (18 total). P(W|A) = $9/18 = 1/2$.
Total Probability $P(W) = P(W|T_W)P(T_W) + P(W|T_B)P(T_B)$.
$P(W) = \frac{5}{9} \times \frac{3}{5} + \frac{1}{2} \times \frac{2}{5} = \frac{1}{3} + \frac{1}{5} = \frac{8}{15}$.
$p=8, q=15$. $\gcd(8,15)=1$.
$p+q = 8+15 = 23$.