Probability Diagrams (Tree & Venn Diagrams)

Revise Probability Diagrams (Tree & Venn Diagrams) for Mathematics 0580 (O Level) — revision notes and instant AI marking.

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📋 What You'll Learn

①

Two-Way Tables — organize data by two characteristics and find probabilities

②

Venn Diagrams — show overlapping sets and calculate probabilities from regions

③

Tree Diagrams — track sequences of events and conditional probabilities

④

Combined Probabilities — use "and/or" rules for multiple events

1️⃣ Two-Way Tables

What is a two-way table?

A two-way table is a table that compares two different characteristics at the same time. For example, you might compare age groups (Year 12 vs Year 13) with language choices (Spanish vs German).

Think of it like a spreadsheet where:

The rows represent one characteristic (e.g. year groups)
The columns represent another characteristic (e.g. languages)
Each cell shows how many items have both characteristics

How to find probabilities from a two-way table

Step 1: Add the totals for each row and column.

Step 2: Add an overall total in the bottom-right corner (this should equal the sum of either the row totals or the column totals).

Step 3: To find a probability, use:

P(outcome) = Number with that outcome / Total number

Important: If you're selecting from a specific category (not the whole group), the denominator is that category's total, not the overall total.

Worked Example

A college has 55 students in two year groups (Year 12 and Year 13) choosing between Spanish and German.

Year Group	Spanish	German	Total
Year 12	15	10	25
Year 13	5	25	30
Total	20	35	55

Find P(student studies Spanish and is in Year 12):

Looking at the intersection of "Year 12" row and "Spanish" column: 15 students.

P(Spanish ∩ Year 12) = 15/55 = 3/11

Find P(student is in Year 13 | they study Spanish):

Now we only care about students who study Spanish (the "Spanish" column total = 20). Of these, 5 are in Year 13.

P(Year 13 | Spanish) = 5/20 = 1/4

Notice: the denominator changed because we restricted our focus to one column.

Common Mistake

Practice Question 1

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Also in the full note

2️⃣ Probabilities from Venn Diagrams
3️⃣ Probability Tree Diagrams
4️⃣ Combined Probabilities
✅ Concepts Checklist
🧠 What to Memorise
⚡ Exam Tips & Common Traps
📌 Final Summary
How to find probabilities from Venn diagrams

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Probability Diagrams (Tree & Venn Diagrams)