Sample space and probabilities of single-stage 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.

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.

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.

When outcomes are not equally likely

Not every chance experiment has equally likely outcomes. Draw a single marble from a bag with different numbers of each colour, and the colours have different probabilities. Here each probability is simply the count of that colour over the total number of marbles, written in lowest terms.

From probability to a prediction

Once you know the probability of an outcome, you can predict how often it should appear over a set number of trials. The expected number of occurrences is the probability times the number of trials. If the probability is one third and you run 30 trials, you expect about 10 of them.

The probability of an event

An event is a chosen part of the sample space, such as rolling an even number on a die. To find its probability, count the favourable outcomes and divide by the total number of outcomes. For a fair six-sided die that total is 6, and the answer is written in lowest terms.

Teaching tip: real coins, dice and bags of coloured counters make probability concrete. Before rolling a die, ask the student to list the sample space and give the probability of each face. Connecting the physical object to the written sample space and fractions grounds the abstract ideas in something they can hold and see.

Reinforce that probabilities run from 0 to 1 and that the outcomes must total 1. If a student suggests a probability greater than 1 or a set that does not sum to 1, it signals a miscounted sample space. Checking the total is a quick and powerful way to catch errors.