# Geometric probability splitting

So, this question has two parts... It's not homework, just something I wanted to calculate, but don't know how to for sure.

So, given an item with a value of X, it has a 50% chance to split/double, so now we have two items, each with value X. They each have separate, and half the chance to split as their parent object, so each one has in this case 25% chance to split. How can I calculate the average number of items, given that the initial chance to split is instead Y, and assuming it can theoretically split forever, and if the splits is also capped to some number N. If somebody could explain exactly HOW they get their answer, that would be a lot more useful :). Maybe format from 1 to N, where N could be infinity?

Also, I have no idea what to tag this thing.

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I figured out the answer myself.... anyway, for this example, we can find the formula by writing out a few terms, we get 1,1.5,1.1815.

The terms become 1/2^1, 1/2^2, 1/2^4,1/2^11, so the formula is sum 1/2^(n*(n+1)/2+1), n=0 to infinity.

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