p-Values: What They Say (and Don't)

Hypothesis testing works like a trial. The null hypothesis is 'innocent until proven guilty' — the boring claim: this drug does nothing, this coin is fair, the groups don't differ.

Then you ask: if the null were true, how surprising would my data be? A fair coin landing heads 9 times in 10 is surprising (p ≈ 0.01). The data made the null look implausible — so we reject it.

The p-value is exactly that: the probability of seeing results at least this extreme if nothing were going on. Small p → surprising under 'nothing's happening' → evidence something is.