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3d6 Exploding — Master Table and Modifier Ladder
Every major test in C.A.T.S. rolls 3d6, exploding: each 6 is re-rolled and added, recursively. Roll-over — meet or beat the Target Number.
Results are unbounded. There is no maximum roll.
Master probability table
+1 worth is how many percentage points of hit chance one point of modifier buys at that Target Number. This is the number every environmental effect converts through.
| Target Number | Chance to succeed | +1 worth |
|---|---|---|
| 3 | 100% | 0.5 pts |
| 4 | 99.5% | 1.4 pts |
| 5 | 98.1% | 2.8 pts |
| 6 | 95.4% | 4.6 pts |
| 7 | 90.7% | 6.9 pts |
| 8 | 83.8% | 8.3 pts |
| 9 | 75.5% | 9 pts |
| 10 | 66.4% | 9 pts |
| 11 | 57.4% | 8.3 pts |
| 12 | 49.1% | 6.9 pts |
| 13 | 42.1% | 6.2 pts |
| 14 | 35.9% | 5.6 pts |
| 15 | 30.3% | 4.9 pts |
| 16 | 25.4% | 4.4 pts |
| 17 | 21% | 3.9 pts |
| 18 | 17.1% | 3.1 pts |
| 19 | 14% | 2.5 pts |
| 20 | 11.4% | 2.1 pts |
| 21 | 9.3% | 1.7 pts |
| 22 | 7.6% | 1.5 pts |
| 23 | 6.2% | 1.3 pts |
| 24 | 4.9% | 1 pts |
| 25 | 3.9% | 0.8 pts |
| 26 | 3.1% | 0.6 pts |
| 27 | 2.5% | 0.5 pts |
| 28 | 2% | 0.4 pts |
| 29 | 1.6% | 0.4 pts |
| 30 | 1.2% | 0.3 pts |
| 31 | 1% | 0.2 pts |
| 32 | 0.8% | 0.2 pts |
| 33 | 0.6% | 0.1 pts |
| 34 | 0.5% | 0.1 pts |
| 35 | 0.4% | 0.1 pts |
| 36 | 0.3% | 0.1 pts |
What one point is actually worth
| Zone | Target Number | +1 worth | Note |
|---|---|---|---|
| Near-certain | 5 | 2.8 pts | modifiers barely matter |
| Comfortable | 8 | 8.3 pts | modifiers bite hard |
| Median | 11 | 8.3 pts | modifiers bite hard |
| Hard | 15 | 4.9 pts | modifiers behave normally |
| Long odds | 20 | 2.1 pts | modifiers barely matter |
| Desperate | 26 | 0.6 pts | modifiers barely matter |
The practical floor on granularity is therefore about 7 to 8 percentage points near the median, not 5. One point of modifier is the smallest unit the engine has, so no effect can be tuned finer than that. This satisfies the design rule of never going finer than 5% — the engine is physically incapable of it.
The modifier ladder
Calibrated at Target Number 11 (the median, 57.4% base). Each step is the modifier whose effect lands closest to the stated percentage band.
Every environmental model in the game emits a percentage, and this table converts it.
| Effect on hit chance | Modifier | Actual chance at TN 11 | True shift |
|---|---|---|---|
| -40% | -7 | 17.1% | -40.4 pts |
| -35% | -6 | 21% | -36.4 pts |
| -30% | -5 | 25.4% | -32 pts |
| -25% | -4 | 30.3% | -27.1 pts |
| -20% | -3 | 35.9% | -21.5 pts |
| -15% | -2 | 42.1% | -15.3 pts |
| -10% | -1 | 49.1% | -8.3 pts |
| -5% | -1 | 49.1% | -8.3 pts |
| 0% | 0 | 57.4% | 0 pts |
| +5% | +1 | 66.4% | +9 pts |
| +10% | +1 | 66.4% | +9 pts |
| +15% | +2 | 75.5% | +18.1 pts |
| +20% | +2 | 75.5% | +18.1 pts |
| +25% | +3 | 83.8% | +26.4 pts |
| +30% | +4 | 90.7% | +33.3 pts |
| +35% | +4 | 90.7% | +33.3 pts |
| +40% | +6 | 98.1% | +40.7 pts |
Drift warning
This ladder is exact only at Target Number 11. The same modifier is worth more near the median and less in the tails, so a -2 that costs 15 points against an even-odds shot costs far less against a desperate one.
That drift is acceptable and is the normal behaviour of any bell curve. It is recorded here so that modifier values are chosen deliberately rather than assumed constant, and so the Player Assistant can display the true current value.
Depth tiers for hit location
A single exploding d6 partitions into exact tiers of five, separated by the unreachable multiples of 6. Reaching a deeper tier requires an explosion.
| Tier | Results | Explosions | Share |
|---|---|---|---|
| 1 | 1-5 | 0 | 83.33% |
| 2 | 7-11 | 1 | 13.89% |
| 3 | 13-17 | 2 | 2.31% |
| 4 | 19-23 | 3 | 0.39% |