Complementary Sets and Subsets

Complementary Set

The complementary event A' is the set of all elements which do not belong to it. It is often symbolized by A' or Ø A. All sampling points of a population are either in A or in A', and no sample point can be a member of both A and A'.

The sum of the probabilities of the event A and its complementary event A' is one.

In some cases, it is easier to calculate P(A') than to calculate P(A). In such cases we can obtain P(A) by P(A) = 1 - P(A').

Example:

When we toss a coin 5 times and define the event A as "at least one head", it is already a lot of work just to list all the possible outcomes. However, we can easily show that the total number of possible outcomes is 2⁵=32. Thus the probability of each outcome is 1/32. The complementary event A' is no heads and consists only of one sample point: TTTTT. So we can calculate the probability of event A by P(A) = 1 - P(A') = 1- 1/32 = 0.96875.

Subsets

When the sample points of event A are a subset of the sample points of B, then A is said to be contained in B, and is written as A Ì B. Thus, when A Ì B, then the occurrence of A necessarily implies the occurrence of B. One can easily see that the probability P(A) is less than the probability P(B).