A bag contains 4 blue, 3 black and 5 yellow balls. At random two balls are drawn from the bag. The probability that both drawn balls are of same colour is :

Question

Accepted Answer

To determine the probability that both drawn balls are of the same color, we need to consider the possible combinations of drawing two balls of the same color from the bag.

The bag contains:
- 4 blue balls
- 3 black balls
- 5 yellow balls

The total number of balls in the bag is 4 + 3 + 5 = 12 balls.

The total number of ways to draw 2 balls from 12 is given by the combination formula $\binom{n}{r}$, which is $\binom{12}{2}$.

$$
\binom{12}{2} = \frac{12 \times 11}{2 \times 1} = 66
$$

Now, let's calculate the number of favorable outcomes for drawing two balls of the same color:

1. **Two blue balls:**
   - The number of ways to choose 2 blue balls from 4 is $\binom{4}{2}$.
   $$
   \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
   $$

2. **Two black balls:**
   - The number of ways to choose 2 black balls from 3 is $\binom{3}{2}$.
   $$
   \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3
   $$

3. **Two yellow balls:**
   - The number of ways to choose 2 yellow balls from 5 is $\binom{5}{2}$.
   $$
   \binom{5}{2} = \frac{5 \times 4}{2 \times 1} = 10
   $$

Adding these favorable outcomes gives us the total number of ways to draw two balls of the same color:

$$
6 \text{ (blue)} + 3 \text{ (black)} + 10 \text{ (yellow)} = 19
$$

Therefore, the probability that both drawn balls are of the same color is:

$$
\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{19}{66}
$$

This matches Option 1, which is $\frac{19}{66}$.

**Explanation of why other options are incorrect:**

- **Option 2: $\frac{29}{66}$**: This option suggests a higher number of favorable outcomes than actually calculated. The correct calculation shows only 19 favorable outcomes, not 29.

- **Option 3: $\frac{1}{6}$**: This option implies a much lower probability than calculated. The fraction $\frac{1}{6}$ would correspond to approximately 11 favorable outcomes, which does not match our calculation of 19.

- **Option 4: $\frac{17}{66}$**: This option suggests fewer favorable outcomes than calculated. The correct number of favorable outcomes is 19, not 17.

Thus, the correct answer is indeed Option 1: $\frac{19}{66}$.