Question

A bag contains 5 white and 3 black balls; another bag contains 4 white and 5 black balls. From any one of these bags, a single draw of two balls is made. Find the probability that one of them would be white and the other black.

• A. 275/504
• B. 5/18
• C. 5/9
• D. None of these

Hint:

Use total probability: average the “1 white, 1 black” probability from each bag, each weighted by 1/2.

Answer 1

The Correct Option is A

This is a two-stage probability problem: first pick a bag, then draw two balls of different colours from it, so the total probability rule applies directly.

1. Bag 1 has 8 balls; ways to pick any 2 is $\binom{8}{2}=28$; ways to pick 1 white and 1 black is $5 \times 3 = 15$; so $P(\text{different colours} \mid \text{Bag 1}) = \frac{15}{28}$.
2. Bag 2 has 9 balls; ways to pick any 2 is $\binom{9}{2}=36$; ways to pick 1 white and 1 black is $4 \times 5 = 20$; so $P(\text{different colours} \mid \text{Bag 2}) = \frac{20}{36}=\frac{5}{9}$.
3. Since each bag is chosen with probability $\frac{1}{2}$, total probability $= \frac{1}{2}\left(\frac{15}{28}+\frac{5}{9}\right)$.
4. Finding a common denominator of 252: $\frac{15}{28}=\frac{135}{252}$ and $\frac{5}{9}=\frac{140}{252}$, summing to $\frac{275}{252}$.
5. Halving gives $\frac{275}{504}$.

So the correct answer is option A, 275/504.