There are 9 red, 7 white and 4 black balls in a bag. If 2 balls are drawn randomly, then the probability that both balls are of the same colour is :

Question

Accepted Answer

To determine the probability that both balls drawn are of the same color, we need to consider the total number of ways to draw 2 balls from the bag and the number of favorable outcomes where both balls are of the same color.

1. **Total number of ways to draw 2 balls from the bag:**

The total number of balls in the bag is 9 (red) + 7 (white) + 4 (black) = 20 balls.

The number of ways to choose 2 balls from 20 is given by the combination formula:
   $$
   \binom{20}{2} = \frac{20 \times 19}{2 \times 1} = 190
   $$

2. **Number of favorable outcomes:**

- **Both balls are red:** There are 9 red balls, and the number of ways to choose 2 red balls is:
     $$
     \binom{9}{2} = \frac{9 \times 8}{2 \times 1} = 36
     $$

- **Both balls are white:** There are 7 white balls, and the number of ways to choose 2 white balls is:
     $$
     \binom{7}{2} = \frac{7 \times 6}{2 \times 1} = 21
     $$

- **Both balls are black:** There are 4 black balls, and the number of ways to choose 2 black balls is:
     $$
     \binom{4}{2} = \frac{4 \times 3}{2 \times 1} = 6
     $$

Adding these favorable outcomes gives us the total number of ways to draw 2 balls of the same color:
   $$
   36 + 21 + 6 = 63
   $$

3. **Probability calculation:**

The probability that both balls drawn are of the same color is the number of favorable outcomes divided by the total number of outcomes:
   $$
   \frac{63}{190}
   $$

Thus, the correct answer is Option 3: $\frac{63}{190}$.

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

- **Option 1: $\frac{36}{190}$** is incorrect because 36 represents only the number of ways to draw 2 red balls, not the total number of favorable outcomes for drawing 2 balls of the same color.

- **Option 2: $\frac{57}{190}$** is incorrect because 57 does not correspond to the sum of the favorable outcomes for drawing 2 balls of the same color.

- **Option 4: $\frac{81}{190}$** is incorrect because 81 exceeds the total number of favorable outcomes for drawing 2 balls of the same color, which is 63.

Therefore, the correct probability that both balls drawn are of the same color is $\frac{63}{190}$, making Option 3 the correct choice.