The expected value (EV) of a Plinko drop is the stake multiplied by (RTP − 1). With a 99% RTP, every unit you drop is worth −0.01 units on average. Variance describes how far real results scatter around that average, and in Plinko it's large enough to hide the edge over short sessions, but not long ones.
This guide works out both numbers for three fictional 8-row pay tables and then checks them with a simulation of 10,000 sessions per setting.
The fictional tables we'll use
Fictional pay tables, built for teaching (full derivation in Plinko risk levels and multiplier tables).
| Slot | Probability | Low | Medium | High |
|---|---|---|---|---|
| 0 / 8 | 0.391% | 5.6× | 13× | 29× |
| 1 / 7 | 3.125% | 2.1× | 3× | 4× |
| 2 / 6 | 10.938% | 1.1× | 1.3× | 1.5× |
| 3 / 5 | 21.875% | 1.0× | 0.7× | 0.3× |
| 4 | 27.344% | 0.5× | 0.4× | 0.2× |
RTPs: low 98.98%, medium 98.91%, high 99.06%.
Expected value: the per-drop average
For a 1-unit stake, the payout X is a random multiplier. Its mean is the RTP, so:
EV(net) = RTP − 1
- Low: 0.9898 − 1 = −0.0102 units per drop
- Medium: 0.9891 − 1 = −0.0109
- High: 0.9906 − 1 = −0.0094
Scale it up linearly: 1,000 drops at 1 unit is about −10 units on any of the three tables. That's the "cost" of playing, on average. You never pay exactly this amount in a given session; you pay it in the long run.
Variance: the per-drop spread
Variance is the average squared distance from the mean:
Var(X) = Σ p(k) × (multiplier(k) − RTP)²
The square root is the standard deviation (SD), in units of stake.
Worked calculation: low risk
| Slots | Multiplier | Probability | (m − 0.9898)² | p × (m − RTP)² |
|---|---|---|---|---|
| 0 + 8 | 5.6 | 0.00781 | 21.2539 | 0.1660 |
| 1 + 7 | 2.1 | 0.06250 | 1.2325 | 0.0770 |
| 2 + 6 | 1.1 | 0.21875 | 0.0121 | 0.0027 |
| 3 + 5 | 1.0 | 0.43750 | 0.0001 | 0.0000 |
| 4 | 0.5 | 0.27344 | 0.2399 | 0.0656 |
| Total | 1 | 0.3114 |
SD = √0.3114 ≈ 0.56 units. Repeating this for the other tables gives about 1.24 (medium) and 2.67 (high). Notice how much of low risk's variance comes from the edge row alone: over half, from slots hit less than 1% of the time.
How a session spreads
Over N drops (each unaffected by the others):
- Expected net result = N × (RTP − 1)
- SD of the net result = SD per drop × √N
For 1,000 drops on high risk: expected −9.4 units, SD ≈ 2.67 × 31.6 ≈ 84 units. The swing is roughly nine times the expected loss. That's why short sessions feel random: they are.
The simulation
To check the formulas, we simulated 10,000 sessions for each table, at 100 and at 1,000 one-unit drops each, using fair 50/50 bounces. Results are net units (payouts minus stakes).
| Risk | Drops | Mean | 5th pct | Median | 95th pct | Sessions ahead |
|---|---|---|---|---|---|---|
| Low | 100 | −1.1 | −9.5 | −1.5 | +9.0 | 38.6% |
| Low | 1,000 | −10.0 | −37.8 | −10.6 | +19.9 | 27.8% |
| Medium | 100 | −0.8 | −18.1 | −2.3 | +22.3 | 42.7% |
| Medium | 1,000 | −10.7 | −70.8 | −13.1 | +57.0 | 37.2% |
| High | 100 | −1.1 | −34.9 | −4.4 | +48.1 | 43.8% |
| High | 1,000 | −9.4 | −141.7 | −13.7 | +136.6 | 43.9% |
Simulated figures; a different random seed gives slightly different numbers. The means match the EV formula within simulation noise.
What the simulation shows
- The mean barely depends on risk. About −1 unit per 100 drops and −10 per 1,000, on every table.
- The spread depends hugely on risk. On 1,000 drops, the middle 90% of high-risk sessions runs from about −142 to +137 units. On low risk, it's −38 to +20.
- "Sessions ahead" falls with more drops on low risk (39% → 28%), because the edge grows faster than the swing. On high risk it stays near 44% at 1,000 drops only because the swing is still huge; run enough drops and it falls too.
- The median is below the mean. Most sessions lose a bit more than average; a few rare edge hits pull the average up. That skew is typical of high-risk tables.
Fictional worked session
Fictional example. A player drops 300 balls at 1 unit each on the high-risk table. Expected net: 300 × −0.0094 ≈ −2.8 units. SD: 2.67 × √300 ≈ 46 units. So a result anywhere between about −90 and +90 is unremarkable. If they finish +40, that isn't evidence of skill or a "good board"; it's well inside normal variance.
Checklist: reading your own results
- Compute expected loss first: total staked × (1 − RTP).
- Compute the typical swing: SD per drop × stake × √(number of drops).
- Compare. If the swing is much larger than the expected loss, your result says almost nothing about the game.
- Don't raise stakes to "recover". EV is fixed per unit staked; bigger stakes mean a bigger expected loss.
- Set a loss limit in advance. High-risk tables can lose 100+ units in 1,000 drops in normal play.
Summary
EV per drop is stake × (RTP − 1), slightly negative on every table. Variance, set by the pay table's steepness, decides how far any session strays from that average. Short sessions are dominated by variance; long ones by the edge. No risk level or staking pattern turns a negative EV positive. For the probability groundwork, see how Plinko works; for verifying individual drops, see how provably fair Plinko works.
FAQ
What is the expected value of a Plinko drop?
Stake × (RTP − 1). With a 99% RTP, a 1-unit drop has an expected value of −0.01 units. Over 1,000 one-unit drops that is about −10 units on average.
Why did I finish ahead after a session if EV is negative?
Variance. Over a few hundred drops the swings are much larger than the small average loss. In our simulation, 37–44% of 1,000-drop sessions on medium or high risk finished ahead, but the average session still lost.
Does variance go away with more drops?
The swing grows with the square root of the number of drops, while the expected loss grows in proportion to the number of drops. So the longer you play, the more the edge dominates.
Can a betting system change the expected value?
No. Changing stake sizes changes how much you risk, not the RTP of each drop. Total expected loss is always total amount staked × house edge.