Standard Deviation (Repeated Trials)

The Mathematics of Gambling by Dr. E. O. Thorp

Expectation is the amount you tend to gain or lose on average when you bet. It, however, does not explain the fluctuations from expectation that occur in actual trials. Consider the fair game example mentioned earlier in the chapter. In a series of any length, we have an expectation of 0. In any such series it is possible to be ahead or behind. Your total profit or loss can be shown to have an average deviation from expectation of about sqr. N. Let D = T - E be the difference of deviation between what you actually gain or lose (T), and the expected gain or loss (E). Therefore, for 100 bets, the average deviation from E=0 is about $10 (in fact, the chances are about 68% and that you will be within $10 of even; they are about 96% that you will be within $20 of even). For ten thousand $1 bets it's about $100 and for a million $1 bets it's about $1,000. Table 2-1 shows what happens. For instance, the last line of Table 1-1 says that if we match coins one million times at $1 per bet, our expected gain or loss is zero (a "fair" game). But on average, we will be about $1000 ahead or behind. In fact, we will be between +$1000 and -$1000 about 68% of the time. (For a million $1 bets, the deviation D has approximately a normal probability distribution with mean zero and standard deviation $1000). We call the total of the bets in a series the "action", A. For one series of one million $1 bets, the action is $1,000,000. However, (fifth column) D/A = 0.0001, so the deviation as a percent of the action is very small. And about 68% of the time T/A is between -.001 and +.001 so as a percent of the action the result is very near the expected result of zero. Note that the average size of D, the deviation from the expected result E, grows – contrary to popular belief. However, the average size of the percentage of deviation, D/A, tends to zero, in agreement with a correct version of the "law of averages."