4.4. THE ABSOLUTE EXCESS OF HEADS OVER TAILS 153
0 .07 to calculate thatP[X 1 ]< 0 ≈ 0 .47210, or about 47%. Alternatively,
if we assume that the individual changesXiare binomial random variables
withP[X 1 = 1] =p, thenE[X 1 ] = 2p−1 and Var [X 1 ] = 4p(1−p). We can
use 2p−1 =E[X 1 ] = 0.07 Var [X 1 ] = 0. 07 ·(4p(1−p)) to solve forp. The
result isp= 0.53491.
In either case, this means that any given piece of advice only has to have
a 53% chance of being correct in order to have a perpetual money-making
machine. Compare this with the strategy of using a coin flip to provide the
advice. Since we don’t observe any perpetual money-making machines, we
conclude that any advice about stock picking must be less than 53% reliable
or about the same as flipping a coin.
Now suppose that instead we have a computer algorithm predicting stock
movements for all publicly traded stocks, of which there are about 2,000.
Suppose further that we wish to restrict the chance thatP[Xannual]< 10 −^6 ,
that is 1 chance in a million. Then we can repeat the analysis to show that
the computer algorithm would only need to haveP[X 1 ] < 0 ≈ 0 .49737,
practically indistinguishable from a coin flip, in order to make money. This
provides a statistical argument against the utility of technical analysis for
stock price prediction. Money-making is not sufficient evidence to distinguish
ability in stock-picking from coin-flipping.
Sources
This section is adapted from the article “Tossing a Fair Coin” by Leonard
Lipkin. The discussion of the continuity correction is adapted from Partial
Sums and the Central Limit Theorem in the Virtual Laboratories in Prob-
ability and Statistics. The third example in this section is adapted from a
presentation by Jonathan Kaplan of D.E. Shaw and Co. in summer 2010.
Problems to Work for Understanding
- (a) What is the approximate probability that the number of heads is
within 10 of the number of tails, that is, a difference of 2% or less
of the number of tosses in 500 tosses?
(b) What is the approximate probability that the number of heads is
within 20 of the number of tails, that is, a difference of 4% or less
of the number of tosses in 500 tosses?