GATE CS 2017 Set 2 QGA-5 — General Aptitude

To find the probability that the two chosen socks are of the same colour, we calculate the ratio of the number of favourable outcomes to the total number of possible outcomes.

Step 1: Calculate the total number of outcomes
The total number of socks is: $$\text{Total socks} = 3 \text{ (red)} + 4 \text{ (green)} + 3 \text{ (blue)} = 10$$ The number of ways to choose $2$ socks out of $10$ is: $$\binom{10}{2} = \frac{10 \times 9}{2} = 45$$

Step 2: Calculate the number of favourable outcomes
For the two socks to be of the same colour, they must be either both red, both green, or both blue:

• Ways to choose $2$ red socks: $\binom{3}{2} = 3$
• Ways to choose $2$ green socks: $\binom{4}{2} = 6$
• Ways to choose $2$ blue socks: $\binom{3}{2} = 3$

Step 3: Calculate the probability
$$\text{Probability} = \frac{\text{Favourable outcomes}}{\text{Total outcomes}} = \frac{12}{45} = \frac{4}{15}$$

Therefore, the correct option is D.