I figured this would be a good place for a question around calculating a result.

I am trying to figure out how to calculate some basic information for a Slot Machine with 3 pieces of information, Coin In, Hold % and Win.

Coin In is the money inserted into the slot machine and wagered, Hold % is the amount of wagered money held back by the slot machine as Win.

If I inserted $10 and the hold % was 10%, then the Win on the $10 would be $1 and $9 would be returned to the player which would then be reinserted, $0.90 would be held, $8.10 returned and so on until nothing remained of the original $10.

This reinserting of the ever decreasing money to re-wager is called recycled money and ultimately once the slot machine was in possession of the entire $10 then the value of recycled money inserted would be $10 / 10% =$100.

It's not exactly how slot machines work but for the sake of this example, we'll imagine that it is.

Now for my question.

If I had the following pieces of Information for a period of time:

Coin In - $1,000,000

Hold % - 9%

Win - $90,000

However my desire for this period was to achieve a 10% hold, not 9% and I am curious to determine how much the slot machine would have won if this 10% had been achieved. Using this 10% value I want to recalculate the Coin In (Recycled Coin) and Win values to see how this would affect them. I seem to be running into a result I don't feel is correct but am ultimately here for validation.

Initially I would imagine that 10% of $1,000,000 would be $100,000 for the Win figure but this would not be correct because recycled Coin In would be reduced from $1,000,000 if a hold of 10% had been achieved and not 9%.

So I figure the Coin In figure would drop from $1,000,000 to $900,000 instead using the Win value of ($90,000 / the 10% Hold ) however I feel this is also wrong since It forces an end result of a $90,000 win even though I achieved an extra 1% hold (9% to 10%). Maybe it's because I'm working backwards from a specific result I only end up with the same result and adjusted Coin In and hold% but I feel the answer lies somewhere in-between a $900,000 - $1,000,000 Coin In value and a $90,000 - $100,000 Win value with more involved than just dividing or multiplying combinations of these 3 values for the correct result.

Any ideas?

Thanks in advance