AC9M7P01 · Year 7 · Probability

Sample space and probabilities of single-stage events

ACARA v9 CONTENT DESCRIPTION “identify the sample space for single-stage events; assign probabilities to the outcomes of these events and predict relative frequencies for related events”

Probability is the mathematics of chance, of how likely things are to happen. Whether a coin lands heads, a die shows a six, or a drawn marble is red, probability lets you measure the likelihood precisely rather than just guessing. This year you learn to list all the possible outcomes of a simple chance experiment, assign probabilities to them, and use those probabilities to predict how often outcomes should occur.

The events in this unit are single-stage events, meaning one action with a clear set of outcomes, like a single roll or a single draw. Getting comfortable with these is the foundation for all the probability that follows, including the multi-step situations you will meet later.

Listing the sample space

Every probability question begins with the sample space: the complete list of outcomes that could possibly happen. For a single flip of a coin, the sample space is heads or tails. For one roll of an ordinary die, it is the six numbers 1 to 6. Writing out the sample space carefully matters, because everything else depends on knowing exactly what can happen, and missing an outcome throws off every probability that follows.

The sample space

The sample space lists every possible outcome of a chance experiment.

The sample space is the complete list of outcomes that could happen. For a single roll of a die, the sample space is the six faces, 1 through 6. Listing every possible outcome is always the first step in working out probabilities.

Some sample spaces are small and easy to list, like the two outcomes of a coin, while others take more care. The key is to be systematic, making sure every possible outcome appears exactly once, with none left out and none counted twice. A complete, accurate sample space is the dependable starting point for assigning probabilities.

Assigning probabilities

A probability is a number between 0 and 1 that measures how likely an outcome is, where 0 means impossible and 1 means certain. When all the outcomes are equally likely, as on a fair die or coin, each outcome gets an equal share of the probability. A fair die has six equally likely faces, so each has a probability of one sixth. A fair coin has two, so each has a probability of one half.

Assigning probabilities

Equally likely outcomes each get the same share, and all probabilities total 1.

Each outcome is given a probability between 0 and 1. When outcomes are equally likely, as on a fair die, each gets the same share: one sixth. The probabilities of all outcomes in the sample space always add up to 1.

Two ideas make probabilities trustworthy. First, the probabilities of all the outcomes in a sample space always add up to 1, because something is certain to happen. Second, once you know the probability of an outcome, you can predict its relative frequency: how often it should occur over many trials. If the probability of red is three quarters, then over many draws about three quarters of them should be red. This link between a single probability and the pattern of many repeated trials is what makes probability such a powerful tool for understanding chance, and it is exactly what the next unit explores through experiments and simulations.

Builds on: The Probability Scale (AC9M6P01). That unit placed the chance of an event on a 0 to 1 scale; here you list the whole sample space and give every outcome its own probability.

Quick self-check

1. What is the sample space of a chance experiment?

2. When you flip a fair coin once, the sample space is

3. On a fair six-sided die, what is the probability of rolling a 4?

4. The probabilities of all outcomes in a sample space must add up to

5. A bag has 3 red and 1 blue marble, all equally likely to be drawn. The probability of red is