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In this book the author John Allen Paulos describes, see full example below, how a random stock pick performs worse than following the newsletter, but I don't understand why. I am still convinced the odds should be 10% for both parties. So, assuming this example is correct. What am I missing? Could someone expand upon this?

A contrived but interesting illustration of a self-fulfilling belief involves a tiny investment club with only two investors and ten possible stocks to choose from each week. Let's assume that each week chance smiles at random on one of the ten stocks the investment club is considering and it rises precipitously, while the week's other nine stocks oscillate within a fairly narrow band. George, who believes (correctly in this case) that the movements of stock prices are largely random, selects one of the ten stocks by rolling a die (say an icosehedron—a twenty-sided solid—with two sides for each number). Martha, let's assume, fervently believes in some wacky theory, Q analysis. Her choices are therefore dictated by a weekly Q analysis newsletter that selects one stock of the ten as most likely to break out. Although George and Martha are equally likely to pick the lucky stock each week, the newsletter-selected stock will result in big investor gains more frequently than will any other stock. The reason is simple but easy to miss. Two conditions must be met for a stock to result in big gains for an investor: It must be smiled upon by chance that week and it must be chosen by one of the two investors. Since Martha always picks the newsletter-selected stock, the second condition in her case is always met, so whenever chance happens to favor it, it results in big gains for her. This is not the case with the other stocks. Nine-tenths of the time, chance will smile on one of the stocks that is not newsletter-selected, but chances are George will not have picked that particular one, and so it will seldom result in big gains for him. One must be careful in interpreting this, however. George and Martha have equal chances of pulling down big gains (10 percent), and each stock of the ten has an equal chance of being smiled upon by chance (10 percent), but the newsletter-selected stock will achieve big gains much more often than the randomly selected ones.

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The key is the distinction between "a stock has big gains" and "a stock results in big investor gains". In the second case, the stock must both be smiled upon by chance, and selected by one of the investors.

For example, the probability that a particular stock is chosen by at least one of the two investors is 19%, and the probability that it realizes big gains that week is 10%, so the probability that the stock results in big investor gains is 1.9% (over one year, you would expect this stock to result in big investor gains once).

The probability that the newsletter-recommended stock is chosen by at least one of the two investors is 100% (since Martha always chooses it) and the probability that it realizes big gains that week is still 10%, so the probability that the newsletter stock results in big investor gains is 10% (over one year, you would expect the newsletter stock to result in big investor gains five times).

At the end of the year, Martha and George could look back at their results, and would (in expectation) see that the newsletter stock resulted in big gains for at least one of them around 5 times more often than a randomly selected stock did. Not because the newsletter had any predictive power, but just because the newsletter stock was more likely to be in their portfolio.

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  • $\begingroup$ I think the problem with the quoted passage is precisely that the author meant to describe a situation in which "a stock results in big investor gains" (for a particular set of investors!), but what he actually wrote was that "a stock has big gains". It's right there in the last sentence: "the newsletter-selected stock will achieve big gains". It doesn't say "for George and/or Martha". I think this was a mistake by the author. $\endgroup$ – David K Aug 28 at 11:49
  • $\begingroup$ Even the earlier phrasing, "will result in big investor gains", seems poorly chosen to me. In every week, every stock has a 10% chance to be the stock that results in big gains for some investors. For the newsletter to have an advantage, we have to replace "investor gains" with "gains for George and Martha's investment club." $\endgroup$ – David K Aug 28 at 11:54
  • $\begingroup$ And at the very end it says, "more often than the randomly selected ones." This is false if "the randomly selected ones" are George's random picks. If they are someone else's random picks independent of George and the newsletter, then yes, they are less likely to make profits for George and Martha. But that's not exactly what the passage says. $\endgroup$ – David K Aug 28 at 11:57
  • $\begingroup$ I agree, it's poorly worded and it's not a particularly good illustration even if the wording was better, but I think my explanation is what was intended by the author. $\endgroup$ – Chris Taylor Aug 28 at 12:00
  • $\begingroup$ I agree, it was almost surely meant to be interpreted the way you did. $\endgroup$ – David K Aug 28 at 12:04

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