## 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

The answer is 2.

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:

ANOTHER UPDATE:

An elementary way to compute the expectation:

“Expectaion” is an anagram of “Inexact Poet“

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Expected number of tosses to get N heads in a row « Drawing Blanks11/25/2009 at 7:20 pm

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On the “Son born on a Tuesday” probability puzzle « Drawing Blanks08/30/2010 at 12:10 am