Runs, Again

The opening section of these notes was an attempt to clarify our intuition about sequences of Bernoulli trials and I now close by augmenting that discussion with a different example of a sequence of Bernoulli events by looking at a simple coin-tossing game between two opponents. The first person wins a dollar from the other player if the binary digit is one (indicating a head); otherwise he losses a dollar to his opponent when the digit is zero (tails). Using a fair coin one expects, as a result of the Law of Large Numbers, that the average gain (or loss) after a large number of tosses to tend to zero as n increases.