Slot Machine Math Question

[Redpipe]

Hi guys,
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

 

[Zexralo8]

Hi guys,
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.
I think slot machine math gets way more interesting when you connect it to real casino play and actual money, because every spin is basically a tiny bet chasing a jackpot that almost never shows up. The probabilities might look clean on paper, but in real gambling sessions variance hits hard and bankroll swings feel wild. I’ve been messing around with payout models lately, and while comparing different bonus structures I stumbled across https://ausscasinosanalyzer.com/au/casino-bonuses/skycrown.com while digging into promos and wagering setups, which actually breaks down offers in a pretty straightforward way so you can see how value really stacks up before placing any bets. Whether someone plays blackjack, roulette, poker, or just spins slots for fun, the math behind expected return always decides who keeps the cash long term, and that’s why understanding probability matters more than chasing lucky streaks or hype from flashy game themes alone these days.
Any ideas?

Thanks in advance

You’re basically looking at expected value. That’s the main idea behind almost every slot-machine type math question. What you do is take each possible outcome, multiply it by its probability, and add everything up. That gives you the long-run average gain or loss per play. So if a machine has different payouts, you don’t just look at the big win - you have to include all the small wins and all the losing spins too. The math is just probability × payout for each case, summed together.

 

[zen0Mesh]

The key thing here is to separate two different levels of how slot machines work. There is the micro level which is each individual spin, and there is the macro level which is the financial reporting over a period of time.

On the micro level, every spin is an independent probabilistic event. A slot has a defined RTP which is return to player. The hold percentage is simply 1 minus RTP. So if RTP is 91 percent, the hold is 9 percent. This does not mean the machine literally keeps 9 percent from every spin. It means that over a very large number of spins, the expected value of total losses converges to 9 percent of total wagers.

Mathematically, the expected value of one spin is the sum of probability times payout for each possible outcome.

However, in your example you are not working at the spin level. You are working with aggregated numbers for a period:

Coin In = 1,000,000
Hold = 9 percent
Win = 90,000

These numbers are internally consistent because Hold = Win divided by Coin In.
90,000 divided by 1,000,000 equals 0.09.

Now we introduce your recycled money idea. What you are describing is effectively a geometric series. If a player inserts an amount B and the hold is h, then each cycle the machine returns B times 1 minus h, which is then wagered again. The total turnover becomes:

B + B times 1 minus h + B times 1 minus h squared and so on.

That infinite geometric series sums to:

Coin In = B divided by h.

So total turnover depends directly on the hold percentage. A smaller hold leads to a larger total recycled Coin In. A larger hold reduces total turnover.

Now the important part. You cannot change the hold from 9 percent to 10 percent and recalculate both Coin In and Win at the same time unless you specify which variable remains fixed. The three values are mathematically tied together by:

Hold = Win divided by Coin In.

So let us examine the two logically consistent cases.

Case 1. If Coin In remains 1,000,000 and hold becomes 10 percent, then:

Win = 0.10 times 1,000,000 = 100,000.

Case 2. If total player loss remains 90,000 and hold becomes 10 percent, then we solve:

0.10 = 90,000 divided by Coin In.

Coin In = 900,000.

Both answers are correct depending on the modeling assumption. There is no value in between. You must decide what stays constant.

In real slot operations, hold is a theoretical long run parameter built into the game math. Traffic, bet size, and number of spins are usually what determine Coin In. Hold then determines the long term ratio between Coin In and Win. You can read more about how slot mechanics and RTP work in practice here: https://magyarcasinobonus.com/befizetes-nelkuli-bonusz/

The important takeaway is that hold is a statistical property of the payout distribution over many spins. When you change hold, you change the structure of the geometric turnover relationship, not just one percentage applied to a fixed number.

So the answer to your question is not somewhere between 900,000 and 1,000,000. The result depends entirely on which variable you treat as fixed in your model. Once two variables are chosen, the third is determined algebraically.