Probability: chance you can calculate
Maths · Key Stage 3 · The School
The 0 to 1 scale
Probability runs from 0 (impossible) to 1 (certain), written as a fraction, decimal or percentage. Rolling a 7 on an ordinary die: 0. Rolling a number under 7: 1. Rolling a 3: 1/6. Anything outside 0 to 1 is not a probability at all — a quick sanity check that catches many errors.
Counting the ways
When outcomes are equally likely, probability = (ways it can happen) ÷ (total ways). A bag holds 3 red and 5 blue counters: P(red) = 3/8. The words "equally likely" matter — a drawing pin does not land point-up half the time just because there are two outcomes. For unfair situations you must gather data, not count outcomes.
Everything sums to one
All the probabilities of one experiment add to 1, so P(not A) = 1 − P(A). If rain has probability 0.3, no rain is 0.7. This one line solves a whole class of questions, especially "at least one" problems, where working out the opposite is far quicker than listing every case.
Two events at once, and the space of everything that could happen
One die is easy. Two is where people start guessing, so draw the SAMPLE SPACE instead: a 6 by 6 grid of every pair, 36 outcomes in all. Now questions answer themselves by counting squares. A total of 7 appears on six of them, so P(7) = 6/36 = 1/6 — the most likely total, which is why it is the one games are built around. A total of 12 appears once: 1/36. And "at least one six" is the case where the opposite is far quicker: 25 of the 36 squares have no six at all, so the answer is 1 − 25/36 = 11/36. Counting the squares you do NOT want, then subtracting, turns most hard probability questions into easy ones.