Sample spaces and equally likely outcomes

By the end of this lesson

By the end you can write down the sample space of a simple random experiment, count the outcomes that belong to an event, and use those counts to find a probability, including the probability that the event does not happen.

What probability is measuring

Probability is a number from $0$ to $1$ that measures how likely an event is.

An event with probability $0$ cannot happen. An event with probability $1$ is certain. Everything in between is a matter of degree, and $0.5$ means "just as likely to happen as not".

To compute a probability we need a precise description of what could happen. That description is the sample space.

Definition

Sample space

An experiment is anything with an uncertain result, such as rolling a die. Each time you do it is called a trial.

The sample space, usually written $S$ , is the set of all possible outcomes of the experiment.

Each single outcome is called an elementary event.
An event is a collection of outcomes, that is, a part (subset) of $S$ . Events get capital letters such as $A$ or $E$ .
$P(E)$ means "the probability that event $E$ happens".

Set notation. Curly brackets $\{\ \}$ mean "the set containing". For one roll of a die, $S = \{1, 2, 3, 4, 5, 6\}$ , and the event "roll an even number" is $E = \{2, 4, 6\}$ .

The counting formula

Suppose the experiment has finitely many outcomes and they are all equally likely. Then, for any event $E$ ,

$P(E) = \dfrac{\text{number of outcomes in } E}{\text{number of outcomes in } S}$

The logic is that probability is shared out evenly between the equally likely outcomes, so each one gets $1/n$ where $n$ is the size of $S$ . An event made of $k$ outcomes collects $k$ of those shares.

The equally likely condition is essential. This formula fails if the outcomes are not equally likely. Rolling two dice and adding the scores is the classic trap: the sums $2$ to $12$ are not equally likely, even though the two dice are fair.

Worked example: Example 1: counting a total

Two fair six-sided dice are rolled and the two scores are added. Find the probability that the total is $8$ .

Have a go first?

Check 1 — reading a sample space

CalculateQuestion 1 · Apply

Multiple choiceQuestion 2 · Recall

Answer the 2 questions above to continue.