The easiest way to approach this problem is as follows :

Probability of atleast 1 pair = 1 – ( Probability of No pair at all)

So calculating the Probability of no pair at all is shown below :

Imagine you have a bag containing 10 pairs of socks. Alright so, lets just assign them some names to make this easier. Lets say Sock 1 & Sock 1′ are pairs.

So you’ll have something like this in your bag :

1 2 3 4 5 6 7 8 9 10

1′ 2′ 3′ 4′ 5′ 6′ 7′ 8′ 9′ 10′

Now when you pick up the sock for the 1st first time, you can pick up and sock. It does’nt matter! So lets say you can take any of the possible 20/20 socks

For simplicity sake, lets say you picked up 7.

When you pick up the second sock however, you cannot pick up 7′. So now out of the remaning 19 socks, you can only pick up 18 possible socks. So you have a choice of 18/19. Say on your second try, you picked up sock 3′ .

Now when you pick the 3rd sock, you cannot pick up 7′ or sock 3. So you have a possible pool of 16 possible socks. So it boils down to 16/18.

Similarly, for the fourth try, you’ll have a possible pool of 14/17 socks.

Now multiplying all these we get

Probability of no Pair of Socks = (20 X 18 X 16 X 14)/(20 X 19 X 18 X 17) = 224/323.

So, the probability of getting atleast one pair will be

Probability of getting atleast one pair = 1 – (Probability of getting no pair)

= 1 – (224/323)

= 99/323

Hope the explanation was simple enough to understand. You can drop a comment below if you found it useful! Cheers!