If you have eight pairs of socks, each pair a different color, find the probability that if you randomly lose?

Question

If you have eight pairs of socks, each pair a different color, find the probability that if you randomly lose five socks, the remaining socks form exactly four matching pairs (and three unmatched socks).

Answer 1

First lets figure the odds that the first four socks you lose are from different pairs, but the fifth is one of the unmatched socks.

You start with 16 socks.

You lose one sock. Odds are 16/16 (1) that it comes from one of the 8 pairs. You now have 15 socks left.

You lose a second sock. Odds are 14/15 that it is from one of the remaining 7 pairs. You have 14 socks left.

You lose the third sock. Odds are 12/14 that it is from one of the remaining 6 pairs. You have 13 socks left.

You lose the fourth sock. Odds are 10/13 that is from one of the remaining 5 pairs. You have 12 left.

You lose the fifth sock. Odds are 4/12 that it is one of the 4 unmatched socks.

The odds of all five of the above happening (in that order) are:
1 x 14/15 x 12/14 x 10/13 x (4/12)
(I put the last one in parentheses, to remind us that this is the sock lost from one of the already unmatched socks).


But that's only one way this can happen. It could also be that only the fourth lost sock is one of the already unmatched. In this case, the odds are the same as above, except that with the fourth sock, you need to lose one of the 3 unmatched socks (3/13) and the fifth sock needs to be lost from the now 5 remaining pairs.

1 x 14/15 x 12/14 x (3/13) x 10/12


Or it could be only the third sock:
1 x 14/15 x (2/14) x 12/13 x 10/12

Or only the second sock:
1 x (1/15) x 14/14 x 12/13 x 10/12

Note that in the last case, the 14/14 means that the odds are 1 (certain) that the third sock comes from one of the remaining 7 pairs. This is true if your first two lost socks are from the same pair (leaving exactly 7 paired socks).

So when you see "or" you add the odds together.

So the answer is
1 x 14/15 x 12/14 x 10/13 x (4/12)
+
1 x 14/15 x 12/14 x (3/13) x 10/12
+
1 x 14/15 x (2/14) x 12/13 x 10/12
+
1 x (1/15) x 14/14 x 12/13 x 10/12

I'll let you do the arithmetic.

Answer 2

its right thank u