The complement of an event

Property The complement of an event is all the outcomes in the sample space that are not included in the event. $$ 1 \operatorname{P}(A) = \operatorname{P}(\text{not } A) $$.

Explanation Why calculate the tricky part when you can calculate the easy part and just subtract? To find the odds of not hitting a target, find the probability of hitting it and subtract that from 1. It’s a clever shortcut that saves you from messy calculations and makes you look like a probability wizard!

Examples A circular town (radius 20 miles) has a square park (side 4 miles). The probability a raindrop misses the park is $$ \operatorname{P}(\text{miss}) = 1 \frac{4^2}{\pi(20)^2} = 1 \frac{16}{400\pi} \approx 0.987 $$. A square wall (side 8 ft) has a triangular target (base 3 ft, height 2 ft). The probability of missing it is $$ \operatorname{P}(\text{miss}) = 1 \frac{\frac{1}{2} \cdot 3 \cdot 2}{8^2} = 1 \frac{3}{64} = \frac{61}{64} $$.