Sock Odds, or, Odd Socks
21/01/15
Nine pairs of socks.
First eight picked at random are all different.
What are the odds?
The other day I was emptying the washing machine. I had removed everything except underwear and socks. I began pulling socks out and found I had extracted eight different socks in a row before I found a match. This seemed quite remarkable to me for only nine pairs, and I mentioned it in conversation to my daughter, who was surprised I hadn't conformed to my character and worked out the odds. I will admit the idea had crossed my mind, but I had resisted it as too trivial. I have learned to resist such urges over the years. Now I was challenged, and resolved to find an answer.......
9 pairs of socks
Pick one at random, there is only one, so no chance of a match
What are the odds of the next one being a match?
There are 17 left and only one of them will match so the odds are 1/17,
or... 16/17 of not matching.
You pick one, it doesn't match
What are the odds of the next one making a pair?
There are now 16 left, either one of the two already out will make a match, so the odds are 2/16
or... 14/16 of not matching.
So, we see a trend developing, as per the table below: -
Try# Unpicked Already picked Odds of a match Odds against Cumulative odds against
1 18 0 0 1 1
2 17 1 1/17 16/17 0.94118
3 16 2 2/16 14/16 0.82353
4 15 3 3/15 12/15 0.65882
5 14 4 4/14 10/14 0.47059
6 13 5 5/13 8/13 0.28959
7 12 6 6/12 6/12 0.1448
8 11 7 7/11 4/11 0.05265
9 10 8 8/10 2/10 0.01053
10 9 9 9/9 0 0
So if I have eight out unmatched, then get a match, what are the odds of this happening?
Multiply the odds of failing to match at each successive pick: -
1 * 16/17 * 14/16 * 12/15 * 10/14 * 8/13 * 6/12 * 4/11
= (16*14*12*10*8*6*4) / (17*16*15*14*13*12*11)
= 5,160,960/98,017,920
= 0.0526532291238174
= 1/18.99218749999999
9 pairs of socks
Pick one at random, there is only one, so no chance of a match
What are the odds of the next one being a match?
There are 17 left and only one of them will match so the odds are 1/17,
or... 16/17 of not matching.
You pick one, it doesn't match
What are the odds of the next one making a pair?
There are now 16 left, either one of the two already out will make a match, so the odds are 2/16
or... 14/16 of not matching.
So, we see a trend developing, as per the table below: -
Try# Unpicked Already picked Odds of a match Odds against Cumulative odds against
1 18 0 0 1 1
2 17 1 1/17 16/17 0.94118
3 16 2 2/16 14/16 0.82353
4 15 3 3/15 12/15 0.65882
5 14 4 4/14 10/14 0.47059
6 13 5 5/13 8/13 0.28959
7 12 6 6/12 6/12 0.1448
8 11 7 7/11 4/11 0.05265
9 10 8 8/10 2/10 0.01053
10 9 9 9/9 0 0
So if I have eight out unmatched, then get a match, what are the odds of this happening?
Multiply the odds of failing to match at each successive pick: -
1 * 16/17 * 14/16 * 12/15 * 10/14 * 8/13 * 6/12 * 4/11
= (16*14*12*10*8*6*4) / (17*16*15*14*13*12*11)
= 5,160,960/98,017,920
= 0.0526532291238174
= 1/18.99218749999999
Conclusion
Near enough to One Chance In Nineteen that I will pick eight in a row without a match
I find this much more likely than I would have thought. Roughly once every five months of weekly washes.
If you think this is just too much thought for a trivial subject, I recommend you not click on the link below to see what happens when a bunch of computer programmers approach the problem...
"Pair socks from a pile efficiently?".