Skill: Determine probabilities of complementary events

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 $1$:

$$P(A)+P(A^c)=1$$

Therefore, use:

$$P(A^c)=1-P(A)$$

Example

The probability that it will rain tomorrow is $\frac{3}{10}$. What is the probability that it will not rain?

1. Let $A$ be the event “it rains.” Then:

$$P(A)=\frac{3}{10}$$

2. The complement, $A^c$, is “it does not rain.”

3. Subtract the probability of rain from $1$:

$$P(A^c)=1-\frac{3}{10}$$

4. Write $1$ as $\frac{10}{10}$ and subtract:

$$P(A^c)=\frac{10}{10}-\frac{3}{10}=\frac{7}{10}$$

So, the probability that it will not rain is:

$$\boxed{\frac{7}{10}}$$

Remember, the complement is found by subtracting from $1$—it is not the reciprocal of the original probability.