Calculator
Pick one of the items that rolls a hyperbolic chance, set the stack count, and read the effective probability. The number in brackets on the item card is the value added to the curve, not the chance you end up with.
Effective chance
60.00%
First stack buys 13.04%; the tenth buys only 2.55%. Both are computed as the change in effective chance, not the flat per-stack value.
| Stacks | Effective chance | Gain from the previous listed step |
|---|---|---|
| 1 | 13.04% | — |
| 2 | 23.08% | +10.03% |
| 3 | 31.03% | +7.96% |
| 5 | 42.86% | +11.82% |
| 8 | 54.55% | +11.69% |
| 10 | 60.00% | +5.45% |
| 15 | 69.23% | +9.23% |
| 20 | 75.00% | +5.77% |
| 30 | 81.82% | +6.82% |
| 50 | 88.24% | +6.42% |
Showing Tougher Times. Per-stack value: +15% — the figure the item's own description states.
Two ways an item can stack
Most items in Risk of Rain 2 add their value directly. Two Tri-Tip Daggers bleed for twice what one bleeds for, and ten stack for ten times. That is linear stacking, and it is what the overwhelming majority of the item list does — 139 of the 145 rows in the community table carry a linear curve.
A handful of items behave differently. Instead of adding to the outcome, each stack adds to a denominator, so the effective value follows 1 − 1/(1 + k×n), where k is the per-stack figure and n is the stack count. The result is a curve that climbs steeply at first and then flattens, closing half of whatever gap remains with every doubling of stacks, and never quite arriving at 100%.
Only 6 items use it: Tougher Times, Stun Grenade, Sentient Meat Hook, Tentabauble, Old Guillotine and Elusive Antlers. Four of those six roll a single chance and are the ones this page computes. The other two change a threshold or an interval rather than rolling, so they are listed but not calculated.
Worked example: Tougher Times
Tougher Times states a 15% chance to block incoming damage, and its own description adds that the effect is unaffected by luck. Plugging k = 0.15 into the curve gives a first stack worth 13.04%, and ten stacks worth 60%. Both of those figures match the numbers the community has used for the item, which is the check that the formula on this page is the right one.
| Stacks | Block chance | The stack that got you there bought |
|---|---|---|
| 1 | 13.04% | — |
| 2 | 23.08% | +10.03% |
| 5 | 42.86% | +19.78% |
| 10 | 60.00% | +17.14% |
| 20 | 75.00% | +15.00% |
| 50 | 88.24% | +13.24% |
| 100 | 93.75% | +5.51% |
The third column is the one worth reading. It is the marginal gain, and it is why a wall of Tougher Times looks unimpressive in the numbers and still performs: the point of the item is not the size of any single block but the share of hits that stop being relevant.
Every hyperbolic item
| Item | Per stack | Rolls a chance? | What the curve means for it |
|---|---|---|---|
| Tougher Times15% (+15% per stack) chance to block incoming damage. Unaffected by luck. | +15% | Yes | The canonical hyperbolic item: the first stack is worth far more than the tenth, which is why stacking it looks disappointing on paper and strong in a run. |
| Stun Grenade5% (+5% on stack) chance on hit to stun enemies for 2 seconds. | +5% | Yes | A small per-stack value, so the curve climbs slowly — many stacks are needed before the chance is dependable. |
| Sentient Meat Hook20% (+20% per stack) chance on hit to fire homing hooks at up to 10 (+5 per stack) enemies for 100% TOTAL damage. | +20% | Yes | Highest per-stack value of the six, so it saturates fastest; the target count also rises linearly with stacks. |
| Tentabauble5% (+5% per stack) chance on hit to root enemies for 1s (+1s per stack). Corrupts all Chronobaubles. | +5% | Yes | Rolls the same curve as Stun Grenade, but the duration scales linearly alongside it. |
| Old GuillotineInstantly kill Elite monsters below 13% (+13% per stack) health. | +13% | No | Marked hyperbolic in the source, but it sets a health threshold rather than rolling a chance, so it is listed here and left out of the probability calculator. |
| Elusive AntlersSpawns orbs of energy nearby every 10s (-10% per stack), giving +12% movement speed up to 3 (+3 per stack) times for 12s. | +12% | No | Changes an interval and a movement buff rather than a single roll, so it is listed but not computed. |
Confirmed by official sources — per-stack values are taken from each item's own description, and the curve classification from the community item table's stacking column.
Where the curve changes your build
The practical consequence is that hyperbolic items are worth picking up early and worth picking up repeatedly, but for different reasons. The first copy is the most efficient single item you can take, because it converts a 0% chance into something meaningful for one stack. The tenth is inefficient by comparison and still correct to take, because the alternative to a 60% block chance is not a 60% block chance from anywhere else.
That also explains why the same item reads as overrated and as mandatory depending on who you ask. Measured in marginal gain it is unimpressive past a handful of stacks; measured in the share of incoming damage it removes, the curve keeps paying. Both readings are of the same numbers.
The last two entries in the table are the exceptions worth knowing. Old Guillotine sets a health threshold rather than rolling, so its per-stack figure moves a cutoff for an instant kill instead of a probability. Elusive Antlers shortens an interval, which means more stacks make its orbs appear more often rather than making any single one stronger. Neither behaves like a chance item, and treating them as one is the usual mistake.
When you have a stack count in mind, the item browser is where to check what else is competing for the slot, and the tier list covers which survivors convert that mitigation into the most damage.
Method and gaps
双曲线堆叠的等效概率按 1 − 1/(1 + k·n) 计算,k 为每堆数值、n 为堆数。该式经两个已知数据点交叉印证:Tougher Times 1 层得 13.0%、10 层得 60%,与社区长期公认值一致。
- Old Guillotine 的执行阈值(低于 13% (+13% per stack) 血量)在来源中同样标为双曲线,但它不是单次掷骰型概率,本站不把它纳入概率计算器,只在表中列出。
- Elusive Antlers 同时改动冷却与移速,同理不纳入概率计算。
- Safer Spaces 在来源的堆叠曲线列中未归入双曲线,因此不在此表内。
- 其余 139 件线性道具的每堆数值尚未逐条录入。
Watch the mechanics explained

Woolie
Explains the proc coefficient scale that the calculator on this page is built on
- Defines the proc coefficient as a value attached to every skill, and states that a coefficient of zero never triggers another item or effect.
- Makes the point that a full coefficient does not mean a guaranteed trigger — it means the item fires at its normal rate.
- Explains that halving the coefficient halves the trigger rate, which is the relationship the formula above assumes.
- Frames the topic as a trade between a faster trigger rate and a larger trigger value, and builds a calculator to compare the two.

Cap_
An introduction to proc coefficients for readers meeting the term for the first time
- Opens from the observation that the same item behaves differently on different survivors, which reads as bad luck until you know why.
- Calls the coefficient a mechanic the game never teaches, so the only way to learn it is from outside sources.
- Treats it as the deciding factor in which items are worth investing in on a given survivor.
Frequently asked
- What is hyperbolic stacking in Risk of Rain 2?
- It is the diminishing-returns curve used by probability items. Each stack adds the same nominal percentage to a divisor rather than to the chance itself, so the effective chance follows 1 − 1/(1 + k×n) and approaches 100% without ever reaching it. Of the 145 item rows in the community table, only 6 are marked hyperbolic — the rest add their value linearly.
- Why does the tenth Tougher Times feel worse than the first?
- Because it is. The first stack takes the block chance from 0% to 13.04%, while the tenth moves it from 57.45% to 60.00%. The per-stack number on the item card never changes, but the share of the remaining gap it closes keeps shrinking.
- Does luck affect these chances?
- Not for Tougher Times — its own description states it is unaffected by luck. That is why stacking it is the only way to move the number, and why builds that rely on it treat the item as a threshold rather than a roll.
- How many stacks until a hyperbolic item is reliable?
- Work backwards from the chance you need. Ten stacks of a 15% item sits at 60%; twenty sits at 75%; fifty reaches 88.24%. The curve never reaches 100%, so a hyperbolic item is always a mitigation of risk rather than a guarantee.