
Re: Prob of flipping coin n times, at no time with #h > #t?
Posted:
Feb 6, 2013 12:12 PM


On Wednesday, February 6, 2013 5:42:18 AM UTC8, 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 ht... sequences of n flips, > > comprising a binomial tree (or pascal's triangle), > > with 5050 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.

