Date: Feb 6, 2013 12:12 PM
Author: RGVickson@shaw.ca
Subject: Re: Prob of flipping coin n times, at no time with #h > #t?

On Wednesday, February 6, 2013 5:42:18 AM UTC-8, JohnF wrote:
> What's P_n, the prob of flipping a coin n times,
>
> and at no time ever having more heads than tails?
>
> There are 2^n possible h-t-... sequences of n flips,
>
> comprising a binomial tree (or pascal's triangle),
>
> with 50-50 prob of going left/right at each node.
>
> So, equivalently, how many of those 2^n paths never
>
> cross the "center line" (#h = #t okay after even number
>
> of flips)?
>
> Actual problem's a bit more complicated. For m<=n,
>
> what's P_n,m, the prob that #h - #t <= m at all times?
>
> That is, P_n above is P_n,0 here. Equivalently, how
>
> many of those binomial tree paths never get >m past
>
> the "center line"?
>
> --
>
> John Forkosh ( mailto: j@f.com where j=john and f=forkosh )


Feller, "Introduction to Probability Theory and its Applications, Vol I (Wiley, 1968), Chapter III, page 89, deals with this (and many related) problems. Chapter II deals with the simple random walk S_k = X_1 + X_2 + ... + X_k, where the X_i are iid and X_i = +-1 with prob. 1/2 each.

On page 89 Feller states and proves Theorem 1: "The probability that the maximum of a path of length n equals r >= 0 coincides with the positive member of the pair p(n,r) and p(n,r+1).

Earlier in Chapter he gave the formula p(n,k)= Pr{S_n = k} = C(n,(n+k)/2)/2^n, where C(u,v) denotes the binomial coefficient "u choose v".

The answer to your "<= m" question is the sum of those probabilities for r from 0 to m, plus the probability that the max is < 0. The latter can be obtained from the expression on page 77, which is
P{S_1 > 0, S_2 > 0, ... S_n > 0} = (1/2)* u(2n),
and where u(2j) = C(2j,j)/2^(2j) = P{S_2j = 0}. Note that having all S_i < 0 has the same probability as having all S_i > 0.