Variance explorer
An expected value is the centre of a distribution. This shows the volatility — how widely results scatter around that centre — on a model game whose mathematics are stated in full, from a seed you can reuse.
This is a simulation, not a prediction
What follows runs a deliberately simple model game many times to show how far results scatter around their average. It describes a model. It does not describe you, it cannot forecast a session, and running more repetitions does not make it able to.
Simulated distribution
Press Run simulation to see how widely the model game’s results scatter. Nothing runs until you ask for it.
How this calculation works
This tool repeatedly simulates one attempt at a wagering requirement and reports the distribution of final balances. Its purpose is to make a single point visible: the expected value the other calculators report is an average, and real results scatter around it very widely.
The game being simulated is a two-outcome model, not a real slot. Each stake returns k times the stake with probability p, and nothing otherwise, with k set to r ÷ p so the model game has exactly the RTP you assume. Lowering the hit frequency p makes wins rarer and larger while holding RTP constant — which is what raising volatility means, and why volatility and RTP are independent.
Every run is driven by a seeded generator, so the same seed and inputs always produce exactly the same output. That is what makes a figure quoted from this page checkable by someone else, and it is why the seed is displayed with the result and included in the copied text.
Watch the relationship between the simulated mean and the closed-form expectation. When runs bust often, the simulated mean sits above the closed form — because a run that empties its balance stops wagering and stops losing, while the closed-form model keeps staking past zero. That gap is precisely the effect the deterministic calculator cannot see, and it is why the expected-balance tool describes its figure as a lower bound.
What you need to enter
| Input | Unit | What it means |
|---|---|---|
| Starting Sweeps Coins | SC | The balance each run begins with. |
| Playthrough multiplier | dimensionless | The requirement each run is trying to satisfy. |
| Game contribution | percent | Share of each coin staked that counts toward the requirement. |
| Assumed RTP | percent | The long-run return the model game is constructed to have. |
| Hit frequency | percent | How often the model game pays anything. This is the volatility control: lower means rarer, larger wins at the same RTP. |
| Stake per play | SC | Held flat for every play in a run. |
| Runs | count | Independent repetitions of the whole attempt, up to 2,000. |
| Seed | text | Any text. It fixes the random sequence completely, so the same seed reproduces the same result exactly. |
The formula
Model game: each stake of size b returns b × k with probability p, and 0 otherwise.
k = r ÷ p (chosen so the model game has exactly the assumed RTP)
Variance of return per unit staked = r² × (1 − p) ÷ p
Standard deviation per unit staked = r × √((1 − p) ÷ p)
A run stops when credited wagering reaches R, or when the balance can no longer cover one stake.
What each symbol means
| Symbol | Name | Unit | Definition |
|---|---|---|---|
B | Starting Sweeps Coins | SC | The Sweeps Coin balance the wagering requirement is calculated from. |
m | Playthrough multiplier | dimensionless | How many times the balance must be wagered before it can be redeemed. A 1x requirement means the balance must be staked once. |
c | Game contribution | fraction of 1 (entered as a percentage) | The share of each Sweeps Coin staked that counts toward the requirement. c = 1 means every coin staked counts in full; c = 0.2 means five coins must be staked for one to count. |
r | Assumed return to player | fraction of 1 (entered as a percentage) | The long-run share of total amount staked that a game returns, averaged over an extremely large number of plays. It is an assumption supplied by you, not a measurement of your session. |
p | Hit frequency | fraction of 1 | How often the model game pays anything at all. Lowering p makes wins rarer and larger while holding RTP constant, which is what raising volatility means. |
b | Stake per play | SC | A constant stake, held flat for every play in the run. |
N | Runs | count | How many independent repetitions of the whole attempt are simulated. |
Worked example
Same RTP, different volatility
- Run 100 SC at 5x with 96% RTP and a 25% hit frequency, and the results cluster fairly close to the expected 80 SC.
- Change nothing except the hit frequency, dropping it to 2%. The RTP is unchanged, so the expected balance is still 80 SC.
- But the distribution is transformed: most runs now finish far below 80 SC, and the average is held up by a small number that finish far above it.
- The standard deviation per unit staked is r × √((1 − p) ÷ p), so at p = 0.25 it is about 1.66, and at p = 0.02 it is about 6.7 — four times wider, at exactly the same RTP.
Takeaway. RTP and volatility are independent. Two games returning the same share over the long run can offer completely different experiences over a session, and the published RTP says nothing about which one you are playing.
Assumptions
These conditions have to hold for the result to mean what it says. Where one does not apply to your situation, the figure will be wrong in a direction the calculator cannot know about.
- The model game is a two-outcome game. It is not any real slot. Real slots typically have many outcome tiers and heavier tails than this model, even at the same RTP and hit frequency. Tail heaviness is not implied by (r, p) alone.
- Every play is independent and identically distributed, and the stake is flat.
- Randomness comes from a seeded generator, so the same seed and inputs always produce exactly the same output.
- The simulation runs in ordinary floating-point arithmetic for speed. The deterministic calculators use exact decimal arithmetic; the simulation does not need that precision because its output is a distribution, not a published figure.
Limitations
What this calculation cannot tell you:
- A simulation describes a model, not a person. It cannot tell you what will happen to your balance, and no number of runs makes it able to.
- Percentile figures are estimates from a finite sample and will move slightly with a different seed. They are reported to a deliberately coarse precision for that reason.
- The distribution of a real game depends on its full paytable, which is not an input here and is rarely published in full.
- Increasing the run count reduces sampling noise. It does not make the model game any more like a real one.
Method and version
This page uses variance-simulation v1.0.0. Every figure it produces comes from that method at that version, and the worked examples are pinned to the same version — when a method changes in a way that moves a published number, the example is re-derived at the new version.
| Version | Date | Change |
|---|---|---|
1.0.0 | 2026-08-15 | First published version. |
Full details of how these models were chosen are on the methodology page.
A note on playing
A simulation describes a model, not a person. Seeing a run in the 95th percentile here tells you that such a run exists in the model, not that it is available to you. Do not use this tool to decide that a result is achievable.
An expected value is an average over an enormous number of repetitions. It is not a prediction, a target, or a result you are owed. See responsible play.