Skill: Find the probability of a complementary event

A complementary event is the event that something does not happen. The event and its complement include every possible outcome, so their probabilities add to $1$:

$$P(\text{complement of } A)=1-P(A)$$

Example

A fair number cube has sides numbered $1$ through $6$. What is the probability of not rolling a number greater than $4$?

1. Let event $A$ be rolling a number greater than $4$.

   The possible outcomes are $5$ and $6$, so:

$$P(A)=\frac{2}{6}=\frac{1}{3}$$

2. Use the complement rule:

$$P(\text{not }A)=1-\frac{1}{3}=\frac{2}{3}$$

3. Check the outcomes. The complement of rolling a number greater than $4$ is rolling $4$ or less: $1,2,3,$ or $4$. That is $4$ outcomes out of $6$:

$$\frac{4}{6}=\frac{2}{3}$$

Therefore, the probability of not rolling a number greater than $4$ is $\boxed{\frac{2}{3}}$.