The complement of event A is denoted Ac, and Ac represents all outcomes not

in A. A and Ac are mathematically related:

EXAMPLE: Rolling a die produces a value in the set {1, 2, 3, 4, 5, 6}. This set of all possible outcomes is called the sample space (S) for rolling a die. We often use the sample space to examine S Sample space

the scenario where an event does not occur.

Let D = {2, 3} represent the event that the outcome of a die roll is 2 or 3. Then the complement of D represents all outcomes in our sample space that are not in D, which Ac Complement of outcome A

is denoted by Dc = {1, 4, 5, 6}. That is, Dc is the set of all possible outcomes not already

included in D. Figure 2.9 shows the relationship between D, Dc, and the sample space S.

P(A) + P(A^c)=1, i.e. P(A)=1P(A^c)

In simple examples, computing A or Ac is feasible in a few steps. However, using the

complement can save a lot of time as problems grow in complexity.

Let A represent the event where we roll two dice and their total is less than 12. (a) What does the event Ac represent? (b) DetermineP(Ac)P(Ac)from Table 2.5 on page 74. (c) Determine P(A).¹⁷