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Re: Prob of flipping coin n times, at no time with #h > #t?
Posted:
Feb 14, 2013 1:42 AM
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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 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)?
>
> 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 (p-q)/(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 n-set 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
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