The Complement Rule

In statistics, the complement rule is a theorem that provides a connection between the probability of an event and the probability of the complement of the event in such a way that if we know one of these probabilities, then we automatically know the other.

The complement rule comes in handy when we calculate certain probabilities. Many times the probability of an event is messy or complicated to compute, whereas the probability of its complement is much simpler.

Before we see how the complement rule is used, we will define specifically what this rule is. We begin with a bit of notation. The complement of the event A, consisting of all elements in the sample space S that are not elements of the set A, is denoted by AC.

Statement of the Complement Rule

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The complement rule is stated as "the sum of the probability of an event and the probability of its complement is equal to 1," as expressed by the following equation:

P(A^C) = 1 – P(A)

The following example will show how to use the complement rule. It will become evident that this theorem will both speed up and simplify probability calculations.

Probability Without the Complement Rule

Suppose that we flip eight fair coins. What is the probability that we have at least one head showing? One way to figure this out is to calculate the following probabilities. The denominator of each is explained by the fact that there are 2⁸ = 256 outcomes, each of them equally likely. All of the following use a formula for combinations:

• The probability of flipping exactly one head is C(8,1)/256 = 8/256.
• The probability of flipping exactly two heads is C(8,2)/256 = 28/256.
• The probability of flipping exactly three heads is C(8,3)/256 = 56/256.
• The probability of flipping exactly four heads is C(8,4)/256 = 70/256.
• The probability of flipping exactly five heads is C(8,5)/256 = 56/256.
• The probability of flipping exactly six heads is C(8,6)/256 = 28/256.
• The probability of flipping exactly seven heads is C(8,7)/256 = 8/256.
• The probability of flipping exactly eight heads is C(8,8)/256 = 1/256.

These are mutually exclusive events, so we sum the probabilities together using the appropriate addition rule. This means the probability that we have at least one head is 255 out of 256.

Using the Complement Rule to Simplify Probability Problems

We now calculate the same probability by using the complement rule. The complement of the event “we flip at least one head” is the event “there are no heads.” There is one way for this to occur, giving us the probability of 1/256. We use the complement rule and find that our desired probability is one minus one out of 256, which is equal to 255 out of 256.

This example demonstrates not only the usefulness but also the power of the complement rule. Although there is nothing wrong with our original calculation, it was quite involved and required multiple steps. In contrast, when we used the complement rule for this problem there were not as many steps where calculations could go awry.​