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20-March-2006
Socks, Stocks and Bundles of Trouble, by Victor Niederhoffer

A most poignant and instructive probability problem is to find the number of odd socks that are created when you start losing single socks from a pair. It turns out that despite normal intuitions, as you lose socks, you seem to create odd unmatched pairs to an inordinate extent. This is an example of a Murphy's law phenomenon that leads us all with drawers full of unmatched socks, baskets of unmatched stocks, non matching shirts and ties, old friends who no longer fit the bill for today and related phenomena..

A typical formulation of the problem is this. Suppose you start with 10 pairs of matching socks of different colors - and then you lose 6 individual socks. What are the chances of ending up with just 4 matching pairs versus seven matching pairs? The answer is that you are 100 times as likely to end up with the worst case scenario of 4 matching pairs as the best case scenario of seven matching pairs. In a discussion of this problem Henk Tijms concludes in Understanding Probability that "When things go wrong they really go wrong." An excellent discussion of this problem with a general solution is contained in Odd Socks: A Combinatoric Example of Murphy's Law

The gist is that if you have 20 socks, 10 different pairs, after you lose the first sock, the chances that the second sock lost will break up yet another pair is 18/19 versus only 1 in 19 that the second sock will kindly be from the original pair first broken up. Now after the second sock was drawn there are 18 socks left. Sixteen of them will unkindly lead to a third pair broken up versus only 2 leading to a match with one of the original pairs. After the third sock is drawn, there are 17 socks left. Fourteen of them will unkindly lead to a fourth set broken up versus 3 that lead to a match, etc..

If you lose 6 socks the exact probability of having them each break up a pair is (10 C6/ 20 C6) all times 2 to the sixth, which come to 0.37. This is the same probability as picking six socks from the 20 without once coming up with a pair. A discussion of this and an exact formula for the probability of picking 0, 1, 2, ... 3 pairs leaving you with 4 pairs, 5 pairs, 6 pairs, 7 pairs is contained in the highly poignant odd socks paper referenced above.

Now, is not this the same phenomenon we witness so often in our market activities. We start with a basket of stocks that each have a reason for being in our portfolio and fit in nicely with our requirements. Then an event occurs and leaves us without a good reason for holding the stock. We lose a match. Now another event or reason for holding a stock goes astray. Again we are left with the likelihood that the stocks and reason don't match. Ultimately the only thing we are left with is a drawer full of stocks that have no rhyme or reason in our portfolio and that are just remaining there because of inertia.

Is not the same thing true with respect to many other phenomena like the friends we are left with, or the names in our rolodex? So often the reasons for their being there in the first place are gone with the wind, and when we look at what's remaining we have a motley collection of old codgers, bags, and misfits, that have no possible value for the current.

What other phenomena come to mind in markets and life that leave us all out of kilter because of a probability chain regrettably related to the odd sock problem?

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