Probably Not, 2nd Edition

8Some Gambling Games in Detail

The Basic Coin Flip Game

The basic coin flip game is an excellent vehicle for examining probability calculations, their results, and implications. Remember, the game is simple: tails give you a dollar to your opponent, heads your opponent gives you a dollar. Assume a fair coin, that is, . The expected value (EV) of return after a game is simply

(8.1)

The EV after any number, n, of flips must similarly be zero. This is what we are calling a fair game.

Figure 8.1 shows (the upper and lower bounds of) the 95% confidence interval of these expected winnings versus the number of coin flips.

Figure 8.1 95% Confidence intervals versus number of coin flips, simple coin flip game.

These numbers are the one‐dimensional random walk “couched in coin's clothing.” As in the random walk example, the standard deviation, , n being the number of coin flips. The expected earnings are 0, but the confidence interval gets wider and wider with increased number of coin flips. At n = 50, for example, you are 95% confident of going home with somewhere between 14 dollars won ...