St Petersburg Game: 1M Trials

  The last post gave the rules for a down-sized version of the St Petersburg Lottery with a maximum of n=4 tosses of a coin. Instead of stopping on "tails" one can flip the coin 4 times each time the game is played ignoring the tosses after the first tail. With 4 tosses there are 16 possible outcomes which we can number 0 through 15 and we can use the digits of the corresponding binary numbers to represent the flips with "0" being a "tail" and "1" a "head". The order of the tosses can be read from right to left and the winnings can be determined and are shown in the following image.

The relative frequencies of an initial run of 0, 1, 2, 3 and 4 heads are seen to be 8, 4, 2, 1 and 1 respectively so the probabilities, 1/2, 1/4, 1/8, 1/16 and 1/16, come out right for the game. N=2²⁰ (1M) trials were run and the results shown below are typical.

The mean value for the wins, μ, is about 3 and is approximately equal to the wager, w=3 as expected for a net gain of zero. The statistics for wins and losses shows that the odds against winning anything is 3:1 as predicted. This allows money to be collected to cover the winnings.

  So instead of tossing coins we can spin a wheel with the numbers 0-15 on it once and pay out the winnings in the table each time but we cannot watch them snowball as in la boule de neige in Roulette. A French song, La Boule, is about the boule de neige system for trying to beat the odds in Roulette.