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Premature Optimization is a Prerequisite for Success

Expected number of coin tosses

What is the average number of coin tosses needed to throw a head?
The trial is a series of n tosses the first n-1 yielding tails and the nth yielding a head. The random variable is n with p(n)=(1/2)^n.
So the expectation is by definition sum_(n=1)^inf(n*(1/2)^n).
How do we compute this? Feed it into Alpha  http://www.wolframalpha.com/input/?i=sum_%28n%3D0%29%5Einf%28n*%281%2F2%29%5En%29
But really, how do we sum the series? The only method I know is to take the geometric series
sum_(n=1)^inf(x^n) = x/(1-x)
and differentiate it.
I don’t have a fully rigorous solution for the expected number of tosses to get m heads in a row.
UPDATE:
Expected number of tosses to get N heads in a row:
https://bbzippo.wordpress.com/2009/11/25/expected-number-of-tosses-general/
ANOTHER UPDATE:
An elementary way to compute the expectation:
https://bbzippo.wordpress.com/2011/06/25/expected-number-of-coin-tosses-once-again/
“Expectaion” is an anagram of “Inexact Poet

Written by bbzippo

11/05/2009 at 3:45 pm

Posted in math

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