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JohnF
Posts:
165
Registered:
5/27/08


Re: Prob of flipping coin n times, at no time with #h > #t?
Posted:
Feb 7, 2013 2:12 AM


Ray Vickson <RGVickson@shaw.ca> wrote: > On Wednesday, February 6, 2013 9:12:51 AM UTC8, Ray Vickson wrote: >> 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".
Thanks, Ray. That sounds like my same problem (but what's "iid" mean in the context "X_i are iid" in your first paragraph above?). I take it my answer should be "the positive member of" p(n,0) and p(n,1), where p(n,k) = C(n,(n+k)/2)/2^n. And I'm guessing that means k=0 if n even, and k=1 if n odd. Actually, I think I already had a solution for the odd cases worked out, but much more complicatedlooking than yours: P_{2n+1} = 0.5  sum(i=1...n) { 1/(2i) * 1/(2^(2i)) * C(2i,i+1) } Note that 2n+1 is always an odd (your k=1, I think) case. Numerically, mine gives (haven't programmed yours yet, to check agreement) n  P_n + 0  .5 1  .375 2  .3125 .  ... 4999 .00798 9999 .00564 19999 .00399 slowly going to zero, i.e., you eventually get an extra head (your r>0, I think), but it may take a lot longer than you'd naively guess (because prob goes to zero very slowly). What I still can't get, in closed form, is equating the two expressions, i.e., for your n>2n+1 and then k=1, yours: p(2n+1,k=1) = C(2n+1,n+1)/2^(2n+1) mine: P_{2n+1} = 0.5  sum(i=1...n) { 1/(2i) * 1/(2^(2i)) * C(2i,i+1) } Are those two really equal? (I'll get around to checking numerically)
>> 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_2n > 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.
Thanks again. My cumbersome derivation wasn't general enough to cover this situation.  John Forkosh ( mailto: j@f.com where j=john and f=forkosh )



