Probability of an Event and Its Complement

Take a fair number cube with six faces numbered $1$ to $6$. Fair means that every face is just as likely to land on top as any other face. Outcomes with the same chance are equally likely.

A roll is certain to show one of the $6$ faces, and a certain event has probability $1$. The $6$ equally likely faces share that $1$ equally, so each face gets $1 \div 6 = \frac{1}{6}$.

Now choose an event: "roll a number greater than $4$". The outcomes that make an event happen are its favourable outcomes. Here they are $5$ and $6$. Only one face can land on top, so the probability of the event is the sum of those two shares: $\frac{1}{6} + \frac{1}{6} = \frac{2}{6}$. Dividing the numerator and the denominator by the same nonzero number, here $2$, keeps the value of the fraction, so $P(\text{greater than } 4) = \frac{2}{6} = \frac{1}{3}$.

Look at what $\frac{2}{6}$ compares: $2$ favourable outcomes with $6$ outcomes in all. The same reasoning works whenever every outcome in the sample space is equally likely. The rule below says this in symbols.

Outcomes are not always numbers. A bag holds $12$ marbles of the same size and feel: $2$ red, $7$ blue and $3$ green. You shake the bag and draw one marble without looking. What is $P(\text{blue})$?

First decide which outcomes are equally likely. You cannot see or feel any difference between the marbles, so each of the $12$ marbles is as likely to come out as any other. Use the $12$ individual marbles as your sample space.

The event "blue" happens when you draw any one of the $7$ blue marbles. It has $7$ favourable outcomes out of $12$ equally likely outcomes, so $P(\text{blue}) = \frac{7}{12}$. That is more than one half, which makes sense because more than half of the marbles are blue.

This probability belongs to one draw from the full bag. If a marble comes out and is not put back, the bag holds different marbles, so the equally likely outcomes change and you would have to count again.

Now look at a new event for the same bag: "not blue". It contains every outcome that is not in the event "blue": the $2$ red marbles and the $3$ green marbles. Mathematicians call "not blue" the complement of "blue". The complement of an event is the set of all the outcomes in the sample space that are not in the event.

You can find its probability by counting. In the table, $2 + 3 = 5$ of the $12$ equally likely marbles are not blue, so $P(\text{not blue}) = \frac{5}{12}$.

There is a second way to find $P(\text{not blue})$. A sample space together with the probability of each outcome is a probability model. For the bag, the model is the $12$ marbles, each with probability $\frac{1}{12}$. In every probability model, the probabilities of all the outcomes add up to $1$, because one of the outcomes is certain to happen. The last row of the table shows this total: $\frac{12}{12} = 1$.

Each marble is either blue or not blue, never both. So the events "blue" and "not blue" together contain every outcome exactly once, and their probabilities add up to the same total:

$P(\text{blue}) + P(\text{not blue}) = 1$

Subtracting $P(\text{blue})$ from both sides keeps the equation true. Write $1$ as $\frac{12}{12}$ so that both fractions have the denominator $12$:

$P(\text{not blue}) = 1 - \frac{7}{12} = \frac{12}{12} - \frac{7}{12} = \frac{5}{12}$

Both ways agree.

Here is the idea to keep. When the outcomes in a sample space are equally likely, the probability of a simple event is the number of favourable outcomes divided by the total number of outcomes, and the answer is always from $0$ to $1$. The probabilities in a model add up to $1$, so any event $A$ and its complement share that whole: $P(\text{not } A) = 1 - P(A)$.

Next, you will compare probabilities to decide which event is more likely, and you will use probabilities to make predictions.