I came across this question where I was asked to calculate the probability of drawing 2 identical socks from a drawer containing 24 socks (7 black, 8 blue, and 9 green). When solving this question, I tried tackling it using the counting principle directly as opposed to simply applying the combinatorics formula in hopes to get comfortable with that principle. Though, when I looked up the solution at the end of the book, the answer was completely different, and I am not sure why that is.
What I did was as follows:
First, I noticed that when we draw the first sock, we have 24 different options to choose from and then only 1 in order to match it with the first one we chose. Then, I divided it by the total number of ways we can draw 2 socks out of 24 and got the expression:
$$\frac{24 \cdot 1}{24 \cdot 23} = \frac{1}{23}$$
The solution at the back of the book took care of each case separately (choosing 2 out of 7 blue socks, OR choosing 2 out of 8 blue socks OR choosing 2 out of 9 green socks), which makes sense, but what is wrong with my approach?
Best Answer
Say you take a green sock out of the drawer. Then for the socks to be identical, you need another green sock, but not all the socks are green. Furthermore, each colour has a different number of socks, so the probability of the second sock matching are all different and so you cannot group different colours of socks.
The same argument applies to any colour.