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

Conditional Probability

In one sentence. Probability when you already know something — shrinking worlds, 'given that' questions, and combining branches over changing conditions. Higher Tier.

Probability — one of 3 modules in this strand, and one of 53 in Pearson Edexcel International GCSE Mathematics. Higher Tier only.

This module ends with Thomas Bayes. Learn Conditional Probability, teach it, then defend it against Thomas Bayes.

Where this sits in the specification

Specification 4MA1, 6.3 Probability — conditional probability (Higher). 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 — conditional probability (Higher)

1 · Understand it

What it is

P(B | A) — 'the probability of B given A' — is B's chance measured inside the world where A has already happened. Formula: P(B | A) = P(A and B)/P(A). In Venn terms: restrict to A's circle, and ask what fraction of it is also in B. Without-replacement second draws ARE conditional probabilities.

Why it works

Knowing A happened shrinks the possibility space to A alone — so probabilities renormalise: divide by P(A). That's the entire formula. In a group of 50 students where 15 own a cat and 6 own both a cat and a dog, P(dog | cat) = 6/15 because the 'given cat' world contains just those 15 students, 6 of whom qualify.

When a question is asking for it

The words 'given that…' or 'if it is known that…'; second-stage branches of no-replacement trees; two-way tables and Venn diagrams (the given event names the row/circle you're confined to). Weather-style chains — P(late) when lateness depends on rain — combine conditionals over each condition: P(late) = P(rain)·P(late | rain) + P(no rain)·P(late | no rain).

How to hold it in your head

'Given that' means 'zoom in': throw away every world where the condition failed, and recount inside what's left. The denominator is always the SIZE of the zoomed-in world, never the whole population.

2 · The mistakes that cost marks

  • Using the whole total as the denominator: P(dog | cat) in the example is 6/15, not 6/50. The 'given' shrinks the world.
  • Confusing P(B | A) with P(A | B): the fraction of smokers who are men ≠ the fraction of men who smoke.
  • Treating dependent events as independent — multiplying unconditional probabilities when the second depends on the first.
  • In 'both same colour' questions, forgetting to ADD the colour-by-colour paths after multiplying each.

3 · Worked examples

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

Of 50 students, 22 own a dog, 15 own a cat and 6 own both. A student who owns a CAT is picked at random. What is the probability they also own a dog? And if instead a DOG owner is picked, what is the probability they own a cat?

  1. Given cat: the world is the 15 cat owners, and 6 of them own a dog. P(dog | cat) = 6/15 = 2/5.
  2. Given dog: the world is now the 22 dog owners, and the same 6 own a cat. P(cat | dog) = 6/22 = 3/11.
  3. Same overlap, different 'given' — so a different denominator and a different answer.

Answer: P(dog | cat) = 2/5; P(cat | dog) = 3/11

P(rain) = 0.25. P(late | rain) = 0.6, P(late | no rain) = 0.1. Find P(late).

  1. Two routes to lateness: via rain and via no rain.
  2. 0.25 × 0.6 = 0.15 and 0.75 × 0.1 = 0.075.
  3. P(late) = 0.15 + 0.075 = 0.225.

Answer: 0.225

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 bag contains 4 red and 6 green sweets. One sweet is taken and eaten — it was red. Work out the probability that the NEXT sweet taken is also red. Give your answer as a fraction in simplest form.
  2. Grade 7 — P(A) = 0.3 and P(A and B) = 0.12. Work out P(B | A) — the probability of B given A.
  3. Grade 8 — In a class of 30 students, 18 study French, 14 study Spanish and 7 study both. A student who studies FRENCH is chosen at random. Work out the probability that they also study Spanish. (Type it like 7/18.)
  4. Grade 9 — A bag contains 5 red and 3 blue counters. Two are taken without replacement. Work out the probability that both counters are the SAME colour. Give your answer as a fraction in simplest form.
  5. 9+ — The probability of rain on a given day is 0.3. If it rains, the probability that Maya is late is 0.5; if it does not rain, it is 0.2. One day Maya is late. Work out the probability that it rained that day. Give your answer as a fraction.

Ask the person who made it

Inside Incandio you can put a question to Alan Turing, who answers knowing exactly which module you have just finished: “At Bletchley you weighed evidence — how much more likely one hypothesis became, given each new observation. How did 'the probability of this, given that' become a weapon that helped break Enigma?”

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 Conditional Probability, marked on the spot, that tell you what was missing rather than just what was wrong.
  2. Teach it — Explain how Conditional Probability 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 Conditional Probability against Thomas Bayes — three rounds, marked against descriptors you can read before you begin.
  4. Then the exam — read all 10 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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