Probability of Long Leads

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Consider a game in which a fair coin is tossed repeatedly. When the cumulative number of heads is greater than the cumulative number of tails, heads is in the lead. Heads retains the lead until the cumulative number of tails is greater than the cumulative number of heads. The probability distribution above indicates the probability of one side being in the lead for different percentages of the duration of the game. For example, if the coin is tossed 20 times, the probability that heads will be in the lead during the entire course of the game is 0.176, the same as the probability that it will never be in the lead. Surprisingly, the least likely situation is for the two sides to be tied for time in the lead.

Contributed by: Fiona Maclachlan (March 2011)
Open content licensed under CC BY-NC-SA



For even values of and , the probability that one side is in the lead for out of tosses is given by:

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