Probability

Stage 8 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 the event is impossible; a probability of 1 means it is certain; $\tfrac{1}{2}$ means an even chance.

For equally likely outcomes, the probability of an event is:

$P(\text{event}) = \frac{\text{number of favourable outcomes}}{\text{total number of outcomes}}$

This year that idea splits into two important kinds of probability — theoretical and experimental — and you will combine two events using a sample space diagram. That is what the rest of this guide covers.

Theoretical probability is the probability you work out by thinking, before you do anything. You reason about the situation, count the equally likely outcomes, and apply the formula.

A fair dice has six equally likely faces. You do not need to roll it to know that:

$P(\text{rolling a 4}) = \frac{1}{6}$

Theoretical probability works whenever the outcomes are equally likely — a fair coin, a fair dice, a well-shuffled pack. It tells you what should happen in the long run.

But the real world does not always match the theory exactly. Flip a fair coin 10 times and you might get 7 heads, not 5. That gap is why we also need experimental probability, the next idea.

Experimental probability is found the opposite way — by doing the experiment and counting what really happens. Each go is called a trial.

$P(\text{event}) = \frac{\text{number of times the event happened}}{\text{total number of trials}}$

Suppose you spin a spinner 50 times and it lands on red 18 times. The experimental probability of red is $\tfrac{18}{50} = 0.36$.

Here is the powerful idea: the more trials you do, the closer experimental probability gets to theoretical probability. Ten coin flips might be uneven, but a thousand flips will be very close to half heads.

Experimental probability is also the only way to estimate the chance of something biased — a weighted dice has no equally likely outcomes, so you must test it to find out how it behaves.

When two things happen together — rolling two dice, spinning a spinner and flipping a coin — a sample space diagram lists every possible combined outcome in a neat grid.

One event runs along the top, the other down the side, and each cell shows one combined outcome.

The grid has $6 \times 6 = 36$ cells, so there are 36 equally likely outcomes. Counting the highlighted cells, 6 of them give a total of 7. So:

$P(\text{total} = 7) = \frac{6}{36} = \frac{1}{6}$

A sample space diagram guarantees you find every outcome — none missed, none repeated — which makes the probability easy and reliable to calculate.

Two outcomes are mutually exclusive if they cannot both happen at the same time. When you roll a dice, "getting a 2" and "getting a 5" are mutually exclusive — one roll cannot be both.

When outcomes are mutually exclusive, the probability that one or the other happens is found by adding their probabilities:

$P(2 \text{ or } 5) = P(2) + P(5) = \tfrac{1}{6} + \tfrac{1}{6} = \tfrac{1}{3}$

This connects to a rule you already met: the probabilities of all the possible outcomes — which are all mutually exclusive — must add up to 1. That gives a handy shortcut. If $P(\text{rain}) = 0.3$, then since "rain" and "no rain" are mutually exclusive and cover everything:

$P(\text{no rain}) = 1 - 0.3 = 0.7$

This is the probability of an event not happening — one minus the probability that it does.

These ideas set you up for plenty of maths ahead — and for thinking clearly about the world:

• Collecting and organising data — experimental probability comes straight from trials recorded in frequency tables, and two-way tables make finding combined probabilities easy.
• Fractions, decimals and percentages — every probability is written in one of these forms, so this topic gives them a real purpose.
• Later probability work — you will go on to combine more events and compare expected results with what actually happens.
• In everyday life, probability shapes weather forecasts, games, insurance, sports predictions and sensible decision-making.

Whenever you hear "there is a 70% chance of rain" or "the dice seems unfair", you now know exactly how someone would test, calculate and reason their way to that statement.