In a Coin Tossing game, how is the probability of a lead change affected by the size of the lead that one side has - taking into account the number of coin flips remaining in the game?

I have asked for help at Statcrunch.com, Hyperstats.com and Wolfram.com without success.

Given a lead at the Halfway Point in a Coin-tossing game of 20, 100, or 1,000 flips, Feller said that the probability of their being no further Lead Changes was roughly 50%.

I have been trying to explore out how the Probability of a Lead Change is affected by the size of the Lead.

So, I manually conducted 20 flip coin games, and collected a sample of 50 games where there was no lead change.

I looked at 39 Leads which had reached a height of 5, and found that 43% finished the game at a higher level than 5, 33% finished at exactly 5, and 23% finished at a lower level than 5.

However, this does not really give me what I want, because I did not record the size of leads in all the other games where there one or more lead changes.

I would like to be able to make a probability statement about the chance of a lead of Size X evaporating into a Lead Change with N number of Coin Flips remaining in a game.

I thought, "Simple. For lead size of 5, just find out the probability of 6 or more heads occurring in a sample of N coinflips. There are all sorts of calculators for that."

However, those calculators fail to take into account whether there were any lead changes during the samples of games of N size that are generated.

So, for example, starting at a Lead of 5 Heads, the next 6 flips might be Tails, which would cause the lead to evaporate and a Lead Change to occur.

But, the overall number of Heads in the next 20 flips might end end up equal to the number of Tails due to Lead Changes...

So, I cannot just use the probability of tossing 13 Tails and 7 Tails in a sample of 20 to figure out the probability of a Lead Size of 5 disappearing given 20 more flips.

I really do not have the time, nor the patience, to do any more manual coin flip experiments and I lack the programming skills to write some simulations to answer my questions.

In basic terms, I want to know when the optimum time is to follow the advice "Quit while you are ahead."

Does not the size of your lead affect the optimum time to quit, given N number of flips remaining in your game?

What is the Probability that you will lose your lead versus increase your lead, depending upon the Size of your Lead and the number of flips remaining?


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