Every lesson so far asked about one hypothesis at a time: disease or not, guilty or not. A probability mass function (pmf) does something bigger: it’s an answer sheet that assigns a plausibility number to every possible value of some discrete quantity at once — not “is it 10%?” but “here’s how plausible every rate from 0% to 100% is, all at once, on one sheet.”
The one hard constraint on that answer sheet: the numbers must be non-negative, and they must sum to exactly 1. That’s not a formula to memorize — it’s just what “plausibility distributed across every possibility” means. If you’re 100% sure the true value is somewhere on your list of possibilities, the total plausibility across the whole list can’t be more or less than “certain.”
A four-sided die (faces 1–4) you suspect is weighted gets tossed many times. From the data, you estimate this pmf:
| face | plausibility |
|---|---|
| 1 | 0.1 |
| 2 | 0.3 |
| 3 | ? |
| 4 | 0.2 |
What must P(3) be, given the pmf has to sum to 1 across all four faces?