Using the Complement Rule to Calculate Probabilities

The first step is to identify the sample space, which consists of all the possible outcomes. There are two number cubes, and each number cube has six possible outcomes. Using the Multiplication Principle, we find that there are $6\times 6,$ or $\text{36 }$ total possible outcomes. So, for example, 1-1 represents a 1 rolled on each number cube.

1. ⓐWe need to count the number of ways to roll a sum of 3 or less. These would include the following outcomes: 1-1, 1-2, and 2-1. So there are only three ways to roll a sum of 3 or less. The probability is
   $$\frac{3}{36}=\frac{1}{12}$$
2. ⓑRather than listing all the possibilities, we can use the Complement Rule. Because we have already found the probability of the complement of this event, we can simply subtract that probability from 1 to find the probability that the sum of the numbers rolled is greater than 3.
   $$\begin{array}{l}P(E^{\prime })=1-P(E)\hfill \\ =1-\frac{1}{12}\hfill \\ =\frac{11}{12}\hfill \end{array}$$