Basic Probability

SPEC P1, P2, P3, P4, P5 FOUNDATION + HIGHER

The 0-1 scale, outcomes, theoretical vs experimental probability, and expected frequency

The Probability Scale

Every probability sits on a scale from 0 to 1. Zero means impossible. One means certain. Everything else lies in between.

Key Rule

A probability must be between 0 and 1 inclusive. Any answer outside this range means the calculation has an error — it can be expressed as a fraction, decimal, or percentage, but never negative, never greater than 1.

Outcomes and Events

Before we calculate anything, we need to be precise about language.

Outcome

A single possible result of one trial. Rolling a fair die has six outcomes: 1, 2, 3, 4, 5, 6.

Event

A set of one or more outcomes — usually the thing we want to find the probability of. "Rolling an even number" is an event made up of three outcomes: 2, 4, 6.

Probability

A measure of how likely an event is, expressed as a fraction, decimal, or percentage between 0 and 1.

Theoretical Probability

When every outcome is equally likely, we can calculate probability exactly using a formula.

P(event) = number of favourable outcomestotal number of outcomes

When to use it

Use theoretical probability when the situation is fair — a fair coin, a fair die, a well-shuffled deck of cards, a random draw from a bag. Every outcome has the same chance.

Mark-scheme Language

"P(event) = favourable over total" — examiners want the fraction written down before the simplified answer. Always show the fraction before the decimal.

Theoretical — Worked Examples

1

Rolling a fair die

Find P(rolling a 4).

Favourable outcomes: just one — the face showing 4.

Total outcomes: six — the numbers 1 to 6.

P(4) = 16

2

Drawing from a deck of cards

Find P(drawing a heart) from a standard 52-card deck.

Favourable outcomes: 13 hearts.

Total outcomes: 52 cards.

P(heart) = 1352 = 14 = 0.25

Experimental Probability

Some situations are not fair — a biased coin, a weighted spinner, a drawing pin landing point-up. We can't assume equally likely outcomes, so we run trials and record the data in a frequency table.

Experimental Probability (Relative Frequency)

P(event) = number of times event occurredtotal number of trials

Frequency table of results

A drawing pin is dropped 80 times:OutcomeFrequencyPoint up52Point down28Total80

Calculate & write it three ways

P(point up) = 5280

Probability can be written as:

Fraction: 52/80 (or 13/20)

Decimal: 0.65

Percentage: 65%

When to use it

Use experimental probability when you cannot assume fairness. The word "biased", "weighted", or any real-world trial data is a signal to use relative frequency.

Frequency Trees

A frequency tree records the counts of outcomes across two categories. The numbers on the branches are frequencies (counts), not probabilities. The AQA spec explicitly requires you to "record, describe and analyse the frequency of outcomes using tables and frequency trees."

80 students surveyed: who plays football (F) and/or tennis (T). Check: 20+30+10+20 = 80. ✓

Reading probabilities off the tree

Every probability is (relevant count) ÷ (total count):

P(plays football) = 5080 = 58 = 0.625

P(plays both) = 2080 = 14 = 0.25

P(plays tennis) = 20 + 1080 = 3080 = 38

Common error

Students read branch values as probabilities. On a frequency tree the branches carry counts. Always divide by the appropriate total before writing P.

Theoretical vs Experimental

Both give a probability between 0 and 1 — but they come from different places. Know which to reach for.

TheoreticalExperimental

Assumes outcomes are equally likelyUses data — no assumption of fairness

favourable ÷ totaltimes event occurred ÷ total trials

Fair coin, fair die, fair spinner, random drawBiased coin, weighted die, real-world experiment

Exact valueEstimate — improves with more trials

More Trials = Better Estimate

Experimental probability is only an estimate of the true probability. But there's a rule that makes the estimate more reliable the more trials you run.

Spec Point P5

Empirical (experimental) unbiased samples tend towards the theoretical probability distribution as sample size increases.

Exam Language

If a question asks "how could the estimate be improved?" — the answer is almost always "increase the number of trials".

Probabilities Sum to 1

If you list every possible outcome, their probabilities must add to exactly 1. This is the "sum to 1" rule.

Exhaustive Set

An exhaustive set of outcomes covers every possibility — there is nothing missing. For an exhaustive set, P(all outcomes combined) = 1.

Ex

Biased spinner with four colours

A biased spinner has exactly four colours: red, blue, green, and yellow. Find P(yellow) given:

P(red) = 0.3, P(blue) = 0.25, P(green) = 0.2, P(yellow) = ?

Because the four colours are the only possible outcomes, they are exhaustive — their probabilities sum to 1:

P(yellow) = 1 − (0.3 + 0.25 + 0.2) = 1 − 0.75 = 0.25

Mutually Exclusive Events

Mutually Exclusive

Two events are mutually exclusive if they cannot both happen at once on the same trial.

Mutually Exclusive

On one roll of a fair die:

"Get a 4" and "get a 6" — mutually exclusive but not exhaustive.

"Odd" and "even" — mutually exclusive and exhaustive (covers all 6 outcomes).

NOT Mutually Exclusive

On one draw from a deck:

"Get a heart" and "get a King"

The King of hearts satisfies both — they overlap.

Addition Rule (mutually exclusive only)

P(A or B) = P(A) + P(B)

This shortcut only works when the events cannot both happen. If they overlap, adding probabilities double-counts the overlap.

The Complement Rule

"Not A" is the event that A does not happen. Since A and "not A" cover every possibility (they are exhaustive and mutually exclusive), they sum to 1.

P(not A) = 1 − P(A)

Ex

Worked mini-example

The probability it rains tomorrow is 0.3. Find the probability it does not rain.

P(not rain) = 1 − 0.3 = 0.7

Exam technique

When "not" appears in a probability question, and finding P(A) looks messy, try finding 1 − P(A) instead. It is often much quicker.

Expected Frequency

Probability tells us how likely something is on a single trial. Expected frequency tells us how often we expect it to happen over many trials.

Expected frequency = n × P(event)

In words

Multiply the probability of the event by the number of trials (n). The answer is an expected count — a number, not a probability.

Watch out

Expected frequency is not the probability — it's the probability times the number of trials. Students often forget to multiply and give back the probability itself.

Expected Frequency — Worked Example

Ex

Rolling a fair die 300 times

A fair die is rolled 300 times. How many times do you expect to roll a 4?

M — Method: Expected frequency = n × P(event).

C — Calculate:

Expected = 300 × 16 = 50

E — Evaluate: We expect 50 rolls of a 4. Note this is a count, not a probability.

Sense-check

Expected frequency must be between 0 and n. For 300 trials, any answer outside 0 to 300 is wrong. Here 50 sits comfortably in range — a good sign.

MCE Practice Question

A biased coin is flipped 200 times. It lands on heads 130 times.

(a) Estimate the probability that the coin lands on heads on the next flip. [1 mark]

(b) Estimate the number of heads if the coin is flipped 500 times. [2 marks]

Total: [3 marks]

M

The coin is biased, so we use relative frequency. Expected frequency = n × P(event).

C

(a) P(heads) = 130200 = 0.65

(b) Expected heads in 500 flips = 500 × 0.65 = 325

E

(a) P(heads) ≈ 0.65, (b) expected heads ≈ 325.

Examiner Alert — Probability Out of Range

Common exam error

Students produce answers like P = 1.2, P = 3/2, or P = -0.1.

Every probability must be between 0 and 1 inclusive. Any answer outside this range is mathematically impossible and signals an error in your method.

What to do

If your answer is outside 0 to 1, stop and check:

Did you swap the numerator and denominator?

Did you add probabilities that shouldn't be added?

Did you forget to subtract from 1 when using "not"?

Sense-check saves marks

Out-of-range answers are one of the easiest marks to save in GCSE probability. A five-second sense-check — "is my answer between 0 and 1?" — catches what a rushed method error misses.

Examiner Alert — Probability vs Expected Frequency

Common exam error

Students confuse the probability (a ratio between 0 and 1) with the expected frequency (a count).

ProbabilityExpected Frequency

P(event) — a ration × P(event) — a count

Between 0 and 1Between 0 and n

"Chance of heads is 0.65""Expected heads in 500 flips is 325"

Read the question carefully

Look for the trigger words: "estimate the probability" wants a ratio; "estimate the number" or "how many" wants a count. Write the unit of your answer clearly.

Common Errors Summary

Students often writeCorrect versionWhy it matters

P = 1.2 or P = 3/2Probability must be between 0 and 1Out-of-range answers lose every mark

Expected frequency = probabilityExpected = n × probabilityForgetting to multiply by trials

P(A or B) = P(A) + P(B) alwaysOnly if mutually exclusiveAddition rule has a condition

Probabilities sum to 100Exhaustive probabilities sum to 1Unless expressed as percentages

"The coin is due a head"Each flip is independentThe "gambler's fallacy"

✓ Checkpoint: What You Should Now Be Able to Do

Core Skills

Place events on the 0 to 1 probability scale using the words impossible, unlikely, evens, likely, certain.

Calculate theoretical probability as favourable ÷ total outcomes for fair situations.

Calculate experimental probability (relative frequency) from a frequency table of trial data.

Read probabilities from a frequency tree by dividing the relevant count by the appropriate total.

Choose between theoretical and experimental based on whether fairness can be assumed.

Write probabilities as fractions, decimals, or percentages as required by the question.

Sum-to-1 and Expected Frequency

Use the fact that exhaustive probabilities sum to 1 to find a missing probability.

Apply P(not A) = 1 − P(A).

Identify mutually exclusive events and apply P(A or B) = P(A) + P(B).

Calculate expected frequency as n × P(event).

Next lesson

Lesson 5.02 extends this to listing outcomes systematically using sample space diagrams, two-way tables, and Venn diagrams — covering spec points P6 and P7.