Complementary Events

Every event has an opposite, called its complement — everything else that could happen. The complement of rolling a 6 is not rolling a 6, which means rolling a 1, 2, 3, 4 or 5. Together an event and its complement cover every possible outcome with no overlap, and two things are always true of them. First, exactly one of them must happen, because there is no third option. Second, they can never both happen at once — you cannot roll a 6 and not roll a 6 on the same throw. Those two facts are what make the shortcut in the next step work.

Because an event and its complement cover everything exactly once, their probabilities must add to 1. So P of A plus P of not A equals 1, which rearranges to the rule you will use constantly: P of not A equals 1 minus P of A. If the probability of a six is one sixth, the probability of not a six is 1 minus one sixth, which is five sixths. The rule works in any form. With decimals, a 0.3 chance of rain gives 0.7 of no rain. With percentages, a 15 percent chance of winning gives 85 percent of not winning — just subtract from 100 percent instead of from 1.

The rule earns its keep when the event you want is harder to count than its opposite. Take two dice and the probability the total is not 7. The long way means counting every pair giving 2, 3, 4, 5, 6, 8, 9, 10, 11 or 12 — ten separate counts, adding to 30 out of 36. The complement way is one line: the probability of a 7 is 6 over 36, or one sixth, so the probability of not 7 is 1 minus one sixth, which is five sixths. Same answer, far less work. Watch for the wording that signals this: 'not', 'other than', 'fewer than', and especially 'at least one', whose complement is always 'none'.

The rule runs in both directions. Given that the probability of not being late is 0.85, the probability of being late is 1 minus 0.85, which is 0.15. Given that the probability of not red is three fifths, the probability of red is two fifths. And given a 72 percent chance of no rain, the chance of rain is 28 percent. Whichever way round the question comes, you subtract from 1 — or from 100 percent. And there is a free check built in: add your two answers together and they must give exactly 1. If they do not, one of them is wrong.

Finally, learn the phrases that signal a complement. 'At least one' is the most common, and its complement is 'none at all', usually a single easy case. 'Not a six' flips to 'a six'. 'Other than blue' flips to 'blue'. 'Fails at least once' flips to 'passes every time'. The one that needs care is a boundary like 'more than 2', because more than 2 means 3, 4, 5 and so on, so its complement is '2 or fewer' — and that includes the 2 itself. Getting the boundary wrong shifts your answer by exactly one case, which is just small enough to slip past unnoticed.