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After watching the biggest lead in Superbowl History evaporate, I looked for info in Feller's classic chapter on coin toss leads, but could not find anything about how to calculate the expected distribution of Lead Sizes at any point during a Coin Tossing Game of length n.

It is easy to find out the distribution of Lead Sizes at the end of a game - but what about during the game?

Yes, 0 would be the expected most frequent Lead Size at the end, but there are often many times during a game when the Lead Size is higher than it is at the end of the game.

To simplify the question, I could limit it to asking what is the Distribution of Largest Size Leads for Heads at any point during a Coin Tossing Game of n Tosses? I supposed we could define "Lead Size" as the difference between nHeads and nTails at any specific point during the game.

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The lead is a simple random walk $S_n = \sum_{k=1}^nX_k$ where $X_k=\pm1$ with 50-50 probability, all independent. Let the maximum lead be $S^*_n = \max_{k=1}^n S_n.$

The probability distribution for the maximum lead is related to the first passage distribution as follows: $$ P(S_n^* < s) = P(\tau(s)>n)$$ where $\tau(s) = \min\{k\ge 1\mid S_k=s\}$ since if the maximum is less than $s$ if and only if the first passage occurs after time $n$. By the reflection principle, the first passage time obeys $$P(\tau(s)\le n) = P(S_n=s) + 2P(S_n>s)$$ so we have $$P(S^*_n\ge s) = 1-P(S^*_ns)$$ as the probability that after $n$ rounds, the maximum lead at any point during the game was greater than or equal to $s.$

The probabilities $P(S_n=s)$ and $P(S_n>s)$ are derivable from the binomial distribution, but there's the complication that $s$ must be the same parity as $n$ for $S_n=s$ to be possible. If $n$ and $s$ have the same parity, then we have $$P(S_n = s) = \frac{1}{2^n}{n \choose \frac{s+n}{2}}$$ and $$ P(S_n >s) = \frac{1}{2^n}\sum_{k= \frac{s+n}{2}+1}^n{n \choose k }.$$

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    I think the question is what is the probability that the maximum of all the leads at all times during the game is $x.$ That's not the same as the probability of lead $x$ at any one particular time. – 2017-02-07
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    @DavidK I thought they were asking about the distribution of max leads at any given time as opposed to the end (cause in feller iirc he derives the distribution of $\max S_n$). Anyway, I edited. – 2017-02-07