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.
 A: 
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.
A: The problem with your answer lies in picking the first sock. Think about this: What must the color of the first sock be to get a pair of sock of the same color?
There are many other ways of thinking of this problem. You can first find the probability of picking, say, a pair of red socks, then multiplying that probability by 3 because you can have 3 possible pairs of socks (with the same color).
Another way of solving this is by finding the amount of possibilities with 2 socks of the same color divided by the total amount of possibilities. In this case, the total amount of possibilities is $$6\times5 = 30$$
Because there are 6 possibilities for the first sock and 5 possibilities for the second sock. 
There are 6 possibilities for choosing two socks of the same color. You can have a pair of red, blue or purple socks. The order does not matter and what that means is that say we have two red socks, $r_1 $ and $r_2$. We can pick $r_1 $ then $r_2$ or $r_2$ then $r_1 $. This gives us two possibilities for each color giving us 6 pairs. To calculate the probability, divide 6 by 30:$$\frac{6}{30}=\frac{1}{5}$$
You can also solve this using permutations or combinations.
A: After picking out the first sock there are $5$ socks left in the drawer. 
Exactly one of them has the same color as the sock that was picked out already.
The probability that this sock will be picked out as second is:$$\frac15$$because the socks have equal probability to be picked out.
