Pearson Edexcel International GCSE in Mathematics (Specification A) · 4MA1 · Free · no account needed

Probability Basics

In one sentence. Probability as fair counting: single events, 'not' events, expected frequency and relative frequency — the foundations every tree diagram stands on.

Probability — one of 3 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Both tiers.

This module ends with Blaise Pascal. Learn Probability Basics, teach it, then defend it against Blaise Pascal.

Where this sits in the specification

Specification 4MA1, 6.3 Probability. Incandio is aligned to this specification; it is not published by, endorsed by or affiliated with Pearson, and the sub-topic numbers are given so you can check every module against your own copy.

  • 6.3 — Probability

1 · Understand it

What it is

Probability measures likelihood from 0 (impossible) to 1 (certain). For equally likely outcomes: P = favourable/total. P(not A) = 1 − P(A), because something must happen. Expected frequency = probability × number of trials. Relative frequency (experimental probability) = times it happened ÷ trials, and it estimates the true probability better as trials grow.

Why it works

P(not A) = 1 − P(A) works because A and not-A split all possibility between them. The equally-likely formula is just fair counting — 4 of the 5 equal sections aren't red, so 4/5. Relative frequency converges on truth (the law of large numbers): 10 flips can easily give 7 heads, but 10 000 flips almost never give 7000.

When a question is asking for it

Spinners, dice, cards → count outcomes. 'How many times would you EXPECT…' → probability × trials. 'Is the coin/dice biased?' → compare relative frequency with the theoretical value, respecting how many trials were run.

How to hold it in your head

Probability is a share of certainty: the whole pie is 1, and every outcome takes its slice. 'Not red' is simply the rest of the pie. Expected frequency shares the trials out the same way the pie is shared.

2 · The mistakes that cost marks

  • Probabilities above 1 or given as '1 in 5' odds-style in an answer box. Probabilities are fractions/decimals in [0, 1].
  • Assuming outcomes are equally likely when they aren't (a 'win/lose' game isn't automatically 50-50).
  • The gambler's fallacy: five heads in a row doesn't make tails 'due'. Coins have no memory.
  • Expecting the expected frequency exactly. 45 is the best single prediction of 300 spins at 0.15, not a guarantee.

3 · Worked examples

Laid out the way an examiner wants working shown — every step is a mark.

A fair spinner has 5 equal sections, one red. What is P(not red)?

  1. P(red) = 1/5.
  2. P(not red) = 1 − 1/5 = 4/5.

Answer: 4/5

Two fair dice are rolled. What is the probability the total is 9?

  1. 36 equally likely pairs in total.
  2. Totals of 9: (3,6), (4,5), (5,4), (6,3) — four pairs.
  3. P = 4/36 = 1/9.

Answer: 1/9

4 · Prove it — the five questions

The five questions climb Grade 6 → 9+ and are marked inside Incandio on your own device, against authored accepted forms, with a diagnosis of the wrong answer you actually gave. The answers stay in the app so that the practice is worth doing; the questions themselves are here.

  1. Grade 6 — A fair spinner has 5 equal sections and exactly 1 is red. Work out the probability of NOT landing on red. Give your answer as a fraction.
  2. Grade 7 — Two fair six-sided dice are rolled and their scores added. Work out the probability that the total is 9. Give your answer as a fraction in simplest form.
  3. Grade 8 — The probability that a biased spinner lands on blue is 0.15. The spinner is spun 300 times. How many blues would you expect?
  4. Grade 9 — A coin is flipped 40 times and lands heads 24 times. Work out the relative frequency of heads.
  5. 9+ — P(A) = 0.4 and P(B) = 0.5, and events A and B are mutually exclusive. Work out P(A or B).

Ask the person who made it

Inside Incandio you can put a question to Richard Feynman, who answers knowing exactly which module you have just finished: “You said nobody understands quantum mechanics — yet its probabilities predict experiments to ten decimal places. How should someone learning probability at school think about a universe that deals in chances rather than certainties?”

Think you understand it?

Reading is the start. On Incandio a topic counts as mastered only once you have taught it and defended it — the two things a revision guide cannot ask of you.

Incandio this topic → Prove it. Teach it. Defend it. What does “Incandio it” mean?

  1. Prove it — five questions on Probability Basics, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Probability Basics works — and why — to Ember, an AI pupil who does not have it yet. The method matters more than the vocabulary.
  3. Defend it — Argue Probability Basics against Blaise Pascal — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 9 authored exam-style questions on this unit, each with its command word and its tariff, then have them marked on Incandio with the mistake you actually made named.

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