Complementary Events

Every event has an opposite, and in probability that opposite has a name: the complement. The complement of an event is everything else that could happen. So the complement of rolling a 6 on a die is not rolling a 6, which means rolling a 1, 2, 3, 4 or 5. Notice that between them, the event and its complement cover all six outcomes with nothing left over. Two things are always true. One of them must happen, because there is no third possibility. And they can never both happen at once, because you cannot roll a 6 and not roll a 6 on the same throw.

Because an event and its complement cover everything with no overlap, their two probabilities must add up to 1. Look at the bar. The whole bar stands for 1, meaning every outcome. One sixth of it is the chance of rolling a six, and the other five sixths is the chance of not rolling a six. Together they fill the bar exactly. So the rule is that P of the event plus P of not the event equals 1. Rearranged, that gives the version you will use most: P of not the event equals 1 minus P of the event. And you can always check, because one sixth plus five sixths is six sixths, which is 1.

The rule works whatever form the probability comes in. With fractions you subtract from 1, so if the chance of a six is one sixth, the chance of not a six is 1 minus one sixth, which is five sixths. If the chance of red is three eighths, rewrite the 1 as eight eighths, then eight eighths minus three eighths gives five eighths. With decimals it is the same: a 0.2 chance of rain leaves a 0.8 chance of no rain. With percentages, subtract from 100 percent instead, so a 35 percent chance of being late leaves a 65 percent chance of not being late. Match whichever form the question gives you.

Questions do not always give you the event and ask for the complement. Sometimes it is the other way round, and the good news is that the rule works in both directions. If the probability of not blue is seven tenths, then the probability of blue is 1 minus seven tenths, which is three tenths. If the probability of no rain is 0.9, then the probability of rain is 0.1. If the probability of not being chosen is 85 percent, then the probability of being chosen is 15 percent. Whichever way round the question comes, you subtract from 1, or from 100 percent. Then add your two answers together as a check — they must come to exactly 1.

Here is where the rule really earns its keep. This bag holds 2 red, 3 blue, 4 green and 1 yellow counter, ten in total. Suppose you want the probability of not red. The long way is to count the blue, the green and the yellow, which is 3 plus 4 plus 1, giving 8 out of 10, or four fifths. That means three separate counts and three chances to slip. The complement way is quicker: the probability of red is 2 out of 10, which is one fifth, so the probability of not red is 1 minus one fifth, which is four fifths. Same answer, one count instead of three. Watch for the words not, does not, and other than.