What's the probability of choosing two socks of the same color from three pairs of socks? I encountered a following problem in probability with independent events:

Six individual socks are sitting in your drawer: two red, two blue and two purple. It's dark, so you can't see a thing.
You pick a first sock uniformly in random, and then a second sock from the remaining five. Assume the two choices are independent.
What's the probability you end up with two socks of the same color?

The app from which I harvested the question says the answer is $1/5$.
But in the case of two independent events, shouldn't the answer be $(2/6) \cdot (1/5)$?
Picking the first sock of the two with the same color has the probability of $2/6$, and therefore picking the second sock with the same color has a probability of $(2-1)/(6-1) = 1/5$ respectively. Multiplying the independent events gives: $(2/6) \cdot (1/5)= 0,0666 \ldots$
The puzzle is from the app called "Probability Puzzles" available in Google Play Store.


My thought process explained: Picking the first sock of the two with the same color has the probability of $2/6$, and therefore picking the second sock with the same color has a probability of $(2−1)/(6−1)=1/5$ respectively. Multiplying the independent events gives: $(2/6)⋅(1/5)$

What you said is true.  However, there are three ways of selecting the color, which gives 
$$3 \cdot \frac{2}{6} \cdot \frac{1}{5} = 1 \cdot \frac{1}{5} = \frac{1}{5}$$
Note that the first sock is guaranteed to be of the same color as those chosen.  It is the only second sock that has to match the color of the first sock.
