Probability Scale, Equally Likely Outcomes and Expected Frequency Revision Notes | GCSE AQA Higher Maths

1. What Is Probability and Why Does It Matter?

Every time you flip a coin, roll a dice, or pick a card, you are dealing with chance. Probability gives you a precise language for talking about that chance — not vague words like "probably" or "unlikely", but exact numbers that examiners can mark.

At GCSE, probability questions appear on both calculator and non-calculator papers. The foundations you build here — the scale, the formula, and expected frequency — underpin every probability topic you will meet, from Venn diagrams to tree diagrams.

2. The Probability Scale

Think of the scale as a sliding dial. At the far left sits "impossible" — rolling a 7 on a standard six-sided dice. At the far right sits "certain" — the sun rising tomorrow. Everything else sits somewhere in between, and your job is to pin down exactly where.

The five key positions on the scale are: 0 (impossible), close to 0 (unlikely), $\frac{1}{2}$ or 0.5 (even chance), close to 1 (likely), and 1 (certain). Examiners often ask you to mark a probability on a scale diagram — always use a fraction, decimal, or percentage, never a word alone.

3. Calculating Probability of Equally Likely Outcomes

When every possible outcome has the same chance of occurring, you can calculate probability precisely using a single formula. This is the most common probability calculation at GCSE.

The word "favourable" does not mean good or desirable — it means the outcomes that count as the event happening. If you want the probability of rolling an even number, the favourable outcomes are 2, 4, and 6, giving three favourable outcomes out of six total.

Worked example: Calculating Probability from a Fair Dice

A fair six-sided dice is rolled once. Calculate the probability of rolling a number greater than 4.

1. Outcomes: 1, 2, 3, 4, 5, 6 — A standard dice has six faces, each equally likely. Total outcomes = 6. [M1 — identifying the sample space]
2. Numbers greater than 4: 5, 6 — Only 5 and 6 satisfy the condition 'greater than 4'. Favourable outcomes = 2.
3. $P(\text{greater than 4}) = \frac{2}{6}$ — Substitute: 2 favourable outcomes out of 6 total. [M1 — correct substitution]
4. $P(\text{greater than 4}) = \frac{1}{3}$ — Divide numerator and denominator by 2. Final answer as a simplified fraction. [A1 — correct simplified answer]

Answer: $P(\text{greater than 4}) = \frac{1}{3}$

Notice that the answer stays as a fraction — this is exact form, which is preferred unless the question asks for a decimal or percentage. On Paper 1 (non-calculator), always leave your answer as a simplified fraction.

4. Probabilities of All Outcomes Sum to 1

Here is a property that unlocks a whole family of exam questions: when you list every possible outcome of an experiment, their probabilities must add up to exactly 1. This makes sense — something has to happen, and the total chance of all possibilities combined is certain.

This rule is especially useful when a question gives you some probabilities and asks you to find a missing one. Set up the equation: all known probabilities plus the unknown equals 1, then solve.

Worked example: Finding a Missing Probability

A biased spinner can land on red, blue, or green. $P(\text{red}) = 0.35$ and $P(\text{blue}) = 0.4$. Work out $P(\text{green})$.

1. $P(\text{red}) + P(\text{blue}) + P(\text{green}) = 1$ — All probabilities for a complete set of outcomes must sum to 1. [M1 — using the sum rule]
2. $0.35 + 0.4 + P(\text{green}) = 1$ — Replace the known probabilities with their values.
3. $0.75 + P(\text{green}) = 1$ — $0.35 + 0.4 = 0.75$
4. $P(\text{green}) = 1 - 0.75 = 0.25$ — Subtract 0.75 from both sides. [A1 — correct answer]

Answer: $P(\text{green}) = 0.25$

5. Expected Frequency

Knowing the probability of an event is useful, but often you need to predict how many times it will happen over a series of trials. That prediction is called the expected frequency.

Expected frequency is a prediction, not a guarantee. If you roll a dice 60 times, the expected frequency of rolling a 3 is $\frac{1}{6} \times 60 = 10$. In practice, you might roll a 3 eight times or twelve times — that is normal variation. The expected frequency tells you what to anticipate on average.

Worked example: Calculating Expected Frequency

A fair coin is flipped 200 times. Calculate the expected frequency of getting tails.

1. $P(\text{tails}) = \frac{1}{2}$ — A fair coin has two equally likely outcomes: heads and tails. [M1 — correct probability stated]
2. Number of trials $= 200$ — The coin is flipped 200 times.
3. $\text{Expected frequency} = \frac{1}{2} \times 200$ — Multiply probability by number of trials. [M1 — correct substitution]
4. $\text{Expected frequency} = \frac{1}{2} \times 200 = 100$ — $\frac{1}{2} \times 200 = 100$. [A1 — correct answer]

Answer: Expected frequency of tails = 100

Worked example: Expected Frequency with a Non-Unit Fraction

A fair six-sided dice is rolled 90 times. Calculate the expected frequency of rolling a 5.

1. $P(5) = \frac{1}{6}$ — There is one 5 on a six-sided dice, so the probability is $\frac{1}{6}$.
2. $\text{Expected frequency} = \frac{1}{6} \times 90$ — Multiply probability by number of trials.
3. $\frac{1}{6} \times 90 = \frac{90}{6} = 15$ — $90 \div 6 = 15$. The expected frequency is a whole number here, but it does not have to be.

Answer: Expected frequency of rolling a 5 = 15

7. Exam Technique and Common Mistakes

The probability topics covered in this lesson appear regularly across both calculator and non-calculator papers. The marks are achievable, but specific errors cost students points every year. Here is what to watch for.

These four foundations — the probability scale, the equally likely outcomes formula, the sum-to-one rule, and expected frequency — are the building blocks for every probability question you will face at GCSE. Secure these methods and you have a reliable platform for tackling Venn diagrams, tree diagrams, and combined events with confidence.