Probability — Trees, Venn Diagrams & Set Theory

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Corbett MathsTree diagrams and conditional probability

Corbett MathsVenn diagrams and set notation

GCSE Maths TutorVenn Diagrams & Set Theory | GCSE Maths Tutor

GCSE Maths TutorProbability Trees (Dependent Combined Events) | Conditional Probability | Grade 6+ | GCSE Maths Tutor

Key facts & methods

Probability rules and tree diagrams

P(A and B) = P(A) × P(B) — for independent events.
P(A or B) = P(A) + P(B) − P(A and B) — addition rule.
Mutually exclusive events: P(A and B) = 0. P(A or B) = P(A) + P(B).
Conditional probability: P(A|B) = P(A and B) / P(B). "Probability of A given B has occurred."
Tree diagrams: multiply along branches (AND). Add the ends (OR). Probabilities must sum to 1 at each branch.
Without replacement: second probability changes after first pick.

Venn diagrams and set theory

A ∪ B = A union B (in A or B or both). A ∩ B = A intersection (in both A and B).
A' = complement of A (not in A).
P(A ∪ B) = P(A) + P(B) − P(A ∩ B).
Venn diagram regions: Only A; Only B; A ∩ B; Neither (outside both circles).
All probabilities in a Venn diagram must sum to 1.
Expected frequency = probability × number of trials.

Exam questions — 4 questions · 10 marks · Edexcel 4MA1 style

4 marksTree diagram — without replacement4MA1 style 🔢 Calculator

A bag contains 5 red and 3 blue counters. Two counters are taken without replacement. Calculate (a) P(both red) (b) P(one of each colour). (4 marks)

Hint: (a) Multiply the two branch probabilities. After taking one red, there are 4 red out of 7 remaining. (b) Two routes: R then B, or B then R — multiply each route then add.

+40 XP

4 marksVenn diagram problemGrade 7 Booklet style ✏️ No calculator

In a class of 30 students: 18 study French (F), 12 study Spanish (S), 5 study both. (a) Draw a Venn diagram and fill in all regions. (b) P(student studies French but not Spanish). (c) P(studies neither). (4 marks)

Hint: Fill in the intersection (both) first: 5. Then: only F = 18−5, only S = 12−5. Neither = total − all in circles.

+40 XP

4 marksConditional probability4MA1 style ✏️ No calculator

Two events A and B. P(A) = 0.6, P(B) = 0.5, P(A ∩ B) = 0.3. (a) Find P(A ∪ B). (b) Find P(A|B). (c) Are A and B independent? Explain. (4 marks)

Hint: (a) Addition rule. (b) P(A|B) = P(A∩B)/P(B). (c) A and B are independent if P(A|B) = P(A). Check if this holds.

+40 XP

1 markExpected frequency4MA1 style

A biased coin gives heads with probability 0.35. If flipped 200 times, what is the expected number of heads?

A 35

B 65

C 70

D 100

1 markMutually exclusive events4MA1 style

Events A and B are mutually exclusive. P(A) = 0.3 and P(B) = 0.25. What is P(A or B)?

A 0.075

B 0.55

C 0.45

D 0.475

Probability — Trees, Venn Diagrams & Set Theory

Probability rules and tree diagrams

Venn diagrams and set theory

Module complete! 🎉