Search All of the Math Forum:
Views expressed in these public forums are not endorsed by
NCTM or The Math Forum.



Re: Prob of flipping coin n times, at no time with #h > #t?
Posted:
Feb 14, 2013 1:42 AM


On Wed, 06 Feb 2013 14:16:33 +0000, Robin Chapman wrote: > On 06/02/2013 13:42, 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)? > > See the ballot theorem: > http://en.wikipedia.org/wiki/Ballot_theorem
Bertrand's ballot theorem addresses the problem "In an election where candidate A receives p votes and candidate B receives q votes with p > q, what is the probability that A will be strictly ahead of B throughout the count?" and gives answer (pq)/(p+q). In the current problem, A must be equal to B or ahead of B throughout the count, so Bertrand's theorem doesn't quite apply and gives wrong answers.
Anyhow, correct answers for the stated question are given by 1/(2^n) times "central binomial coefficients: C(n,floor(n/2)" (see <http://oeis.org/A001405>).
As noted at that link, central binomial coefficients count "the maximal number of subsets of an nset such that no one contains another" (ie max sizes of Sperner families) <http://en.wikipedia.org/wiki/Sperner_family#Sperner.27s_theorem> which probably can be equated to the current problem with a little work.
Central binomial coefficients also count the "number of left factors of Dyck paths, consisting of n steps", where a Dyck path is a "staircase walk from (0,0) to (n,n) that lies strictly below (but may touch) the diagonal y=x" which has an obvious parallel to the current problem. <http://mathworld.wolfram.com/DyckPath.html>
 jiw



