Chance of Meeting a Target Number on One Die

This is classical probability at its most direct: count the faces that succeed and divide by the faces there are. If the target is $t$ and the die has $d$ faces, the successes are $t, t+1, \ldots, d$, and there are $d - t + 1$ of them.

The plus one is the whole difficulty, and it is where nearly every error in this arithmetic comes from. Counting from 11 to 20 on a twenty-sided die gives ten faces, not nine: both ends are included, because the target itself succeeds. The general name for this is the fencepost error — a fence a hundred metres long with posts every ten metres needs eleven posts, not ten — and it is worth checking by hand the first few times. If the answer feels one face short, it is.

Two boundary cases confirm the formula behaves. A target of 1 gives $(d - 1 + 1)/d = 1$: every face clears it, which is right. A target of $d + 1$ gives zero: nothing on the die reaches it, which is also right, and the calculator flags it as an error rather than returning a bare zero, because a target above the die is far more often a typing mistake than a deliberate impossibility.

What the formula assumes is a FAIR die, and that assumption is doing quiet work. Every face must be equally likely. Cheap injection-moulded dice with hollowed pips are measurably not fair, casino dice are made to tolerances precisely because it matters, and a die that has been dropped on a corner may no longer be the die it was. None of this changes the arithmetic; it changes whether the arithmetic applies.

And once again: the die has no memory. Four failures in a row do not improve the fifth attempt by any amount whatever. This is the page where that error does the most damage, because a target roll is usually attempted repeatedly and it is exactly the setting in which "I'm due" feels most reasonable. The chance is $(d - t + 1)/d$ on the first attempt and on the thousandth, unchanged and unchanging. If you want the chance that at least one of several attempts succeeds, that is a different and answerable question — it is one minus the chance they all fail — and it lives on its own page. What it is not, ever, is a rising chance on any individual roll.