expected number of tosses to Win coin toss game.

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steve newman

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I have seen, and understand, many derivations of the expected duration of a coin toss game where you start with $n, and gain or lose $1 with each toss.
The game ends when you have $0, or $N (N>n). But these derivations that i have seen only give the expected duration of such games that include both
outcomes- win $(N-n) or lose $n.

What I would like to see is a derivation of the expected duration of just those games where you win, and the expected duration of just those
games where you lose.

I'd be happy to see the results for those games where the coin is fair (50-50 chance of winning or losing a toss), and those games where the coin is unfair (chance to win = p not = .5)

( I have actually stumbled on the solution for the fair game situatiion, but i would like to see how it is derived.
the solution is - average duration of Lost games is 1/3 x (N^2 - (N-n)^2 ), Average duration of Won games is 1/3 x (N^2-n^2) )
 
found a derivation of the expected duratiion time of Won games.

A lot of people (86)have read this thread, but no one
responded with a reference to a solution.
In the meantime, i have found a short,
elementary derivation of the result-
Expected time to WIN gamblers ruin n,N = 1/3 (N^2 - n^2)
LOSE " 1/3 ( N^2 - (N-n)^2 )
but its a little too long to type up and insert here.

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steve newman said:
I have seen, and understand, many derivations of the expected duration of a coin toss game where you start with $n, and gain or lose $1 with each toss.
The game ends when you have $0, or $N (N>n). But these derivations that i have seen only give the expected duration of such games that include both
outcomes- win $(N-n) or lose $n.

What I would like to see is a derivation of the expected duration of just those games where you win, and the expected duration of just those
games where you lose.

I'd be happy to see the results for those games where the coin is fair (50-50 chance of winning or losing a toss), and those games where the coin is unfair (chance to win = p not = .5)

( I have actually stumbled on the solution for the fair game situatiion, but i would like to see how it is derived.
the solution is - average duration of Lost games is 1/3 x (N^2 - (N-n)^2 ), Average duration of Won games is 1/3 x (N^2-n^2) )
Click to expand...
 
found it.

I have since been able to derive the result.

It's a little too long to type up, but it is short
and elementary.
 
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