Now, we discuss a trick that helps in some problems. Sometimes, instead of finding the probability of an event, it can be easier to find the probability that the event does not occur. Since we know that either an event will occur or not, we know that the probability of the event occurring plus the probability of the event not occurring must equal to 1. More formally, we write:

Probabilities for Complements
Let A be some event. Then, P(A) + P(A does not occur) = 1.

Here's an example where we apply this rule:

Example 1
You are about to roll a fair six-sided dice. What is the probability that you roll a prime number?

Why do we bother with this rule? Sometimes it is actually more convenient to calculate things this way. For example,

Example 2
You are going to roll a fair six-sided dice 10 times in a row. What is the probability that you get a six at least once?

Now, it will often be easier to just directly calculate the probability of a given event. But if that calculation looks absurdly difficult or tedious, take a minute to step back and consider: can I calculate the probability that the event doesn’t occur? Would that be easier? Sometimes that can save you a lot of hassle.

Practice Problems

Question 1
You are rolling a fair spinner with seven sections, numbered 1 through 7. What is the probability that, in 10 spins, you get at least one prime number?

Question 2
One random employee will be selected to win a company-sponsored vacation. There are 500 employees, 300 of which are female. Also, 50 employees are managers, and half of the managers are female. What is the probability that the winner is not a female manager?

Question 3
The weather reporters say that there is a 60% chance of rain for each of the next seven days. If that is true, what is the probability that it rains at some point over the next seven days?