Probability

Stage 9 probability builds on what you already know, so a quick refresher first.

Every probability is a number on a scale from 0 to 1. A probability of 0 means impossible, 1 means certain, and $\tfrac{1}{2}$ means an even chance.

For equally likely outcomes: $P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}.$

Theoretical probability is worked out by reasoning — a fair dice gives $P(4) = \tfrac{1}{6}$ without rolling. Experimental probability is found by doing trials and counting: successes $\div$ total trials.

This year the focus is combined events — two or more events happening together or one after another — and a powerful new tool, the tree diagram. That is what the rest of this guide covers.

A combined event happens when two or more events occur together — flipping a coin and rolling a dice, or picking a counter and then picking another.

The safest way to handle a combined event is to list every possible outcome in an organised way, so none is missed and none is repeated. This is called systematic listing.

For a coin and a dice, list the coin's results in order, then pair each with the dice's $1$ to $6$. That gives $2 \times 6 = 12$ outcomes.

This points to a useful rule: the total number of combined outcomes is found by multiplying the number of outcomes of each event. Two events with $3$ and $4$ outcomes give $3 \times 4 = 12$ combined outcomes. Listing them, or using the diagrams in the next sections, then lets you count the favourable ones and find the probability.

A sample space diagram is a clear grid that lists every combined outcome of two events — one event along the top, the other down the side, with each cell showing one outcome.

The grid has $6 \times 6 = 36$ cells, so there are $36$ equally likely outcomes. To find $P(\text{total} = 7)$, count the highlighted cells — there are $6$ — so: $P(\text{total} = 7) = \frac{6}{36} = \frac{1}{6}.$

A sample space diagram guarantees you find every outcome, which makes the probability reliable. It works best when each event has a small, clear set of outcomes. For longer sequences, the next tool — a tree diagram — is better.

A tree diagram shows a sequence of events as a set of branching paths. Each event adds a new set of branches, and every branch is labelled with its probability.

There are two golden rules for tree diagrams:

• Multiply along the branches. The probability of a whole path is found by multiplying the probabilities on each branch of that path. For Red then Red: $0.3 \times 0.3 = 0.09$.
• Add the separate paths. If more than one path gives the result you want, add their probabilities.

For example, $P(\text{one red and one blue, in any order})$ uses two paths — Red-Blue and Blue-Red: $0.21 + 0.21 = 0.42$.

A handy check: the probabilities on the branches from any single point always add to 1, and all four end probabilities ($0.09 + 0.21 + 0.21 + 0.49$) also add to $1$.

For combined events, you can find probability two ways — by reasoning (theoretical) and by doing trials (experimental) — and it is worth comparing them.

Theoretical: rolling two dice, a sample space diagram shows $P(\text{total} = 7) = \tfrac{6}{36} = \tfrac{1}{6} \approx 0.167$.

Experimental: actually roll two dice many times and count the sevens. With $30$ rolls you might get $7$ sevens, an experimental probability of $\tfrac{7}{30} \approx 0.23$ — close, but not exact.

The key idea, just as for single events: the more trials you do, the closer experimental probability gets to theoretical. A few rolls can look uneven; thousands of rolls settle close to $\tfrac{1}{6}$.

If the experimental probability stays far from the theoretical value even after many trials, it is a strong hint that the dice or spinner is biased — the outcomes are not really equally likely.

These ideas set you up well for the probability ahead — and for thinking clearly about chance:

• Later probability work — you will go on to handle longer sequences and events where one outcome changes the next.
• Fractions, decimals and percentages — every probability is written in one of these forms, so this topic gives them a real purpose.
• Data handling — experimental probability comes straight from trials recorded in frequency tables.
• In everyday life, combined-event reasoning shapes games, weather forecasts, insurance and sensible decision-making.

The two golden rules — multiply along branches, add the paths — are the heart of all the probability you will meet later. Practise them on tree diagrams now, and harder probability will feel familiar when it arrives.