Probability of Same Colour Socks

This question involves calculating the probability of a specific event: selecting two socks of the same color from a mixed collection. We will use combinations and probability principles to find the answer.

First, let's list the number of socks of each color:

• Red socks: 3
• Green socks: 4
• Blue socks: 3

The total number of socks is the sum of all colors: $3 + 4 + 3 = 10$ socks.

We are asked to choose exactly 2 socks from this total of 10.

Total Ways to Choose 2 Socks

To calculate probability, we need the total number of possible outcomes. This is the number of ways to choose any 2 socks from the 10 available socks, irrespective of their color. Since the order of selection doesn't matter, we use combinations.

The formula for combinations is $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where n is the total number of items and k is the number of items to choose.

Here, n = 10 (total socks) and k = 2 (socks to choose).

The total number of ways to choose 2 socks from 10 is:

Total combinations = $\binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10!}{2!8!} = \frac{10 \times 9}{2 \times 1} = 45$.

So, there are 45 different possible pairs of socks that can be selected.

Favourable Ways for Same Colour Socks

Next, we need to count the number of ways to choose 2 socks that are the same color. This can happen in three ways:

1. Choosing 2 red socks.
2. Choosing 2 green socks.
3. Choosing 2 blue socks.

Ways to Choose 2 Red Socks

There are 3 red socks. The number of ways to choose 2 red socks is:

Ways (2 Red) = $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2}{2 \times 1} = 3$.

Ways to Choose 2 Green Socks

There are 4 green socks. The number of ways to choose 2 green socks is:

Ways (2 Green) = $\binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4!}{2!2!} = \frac{4 \times 3}{2 \times 1} = 6$.

Ways to Choose 2 Blue Socks

There are 3 blue socks. The number of ways to choose 2 blue socks is:

Ways (2 Blue) = $\binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3!}{2!1!} = \frac{3 \times 2}{2 \times 1} = 3$.

The total number of favourable outcomes (pairs of the same color) is the sum of these possibilities:

Total favourable ways = Ways (2 Red) + Ways (2 Green) + Ways (2 Blue)

Total favourable ways = $3 + 6 + 3 = 12$.

Final Probability Calculation

The probability of an event is the ratio of favourable outcomes to the total possible outcomes.

Probability (Same Colour Socks) = $\frac{\text{Total favourable ways}}{\text{Total ways to choose 2 socks}}$

Probability = $\frac{12}{45}$.

To simplify the fraction $\frac{12}{45}$, we divide both the numerator and the denominator by their greatest common divisor, which is 3.

Simplified Probability = $\frac{12 \div 3}{45 \div 3} = \frac{4}{15}$.

Thus, the probability of choosing two socks of the same color is $\frac{4}{15}$.