Probability of Combined Events (Copy)

Cheat Sheet: Probability of Combined Events (IGCSE Mathematics 0580 – CORE)
Topic: Sample Space Diagrams, Venn Diagrams, Tree Diagrams (with Replacement)

1. Sample Space Diagrams

• Sample space = all possible outcomes.
• Use to find combined event probabilities by listing all possible outcomes.
• To find probability of an event:

Probability=Number of favorable outcomesTotal number of outcomestext{Probability} = frac{text{Number of favorable outcomes}}{text{Total number of outcomes}}

Example:
Tossing two coins → Sample space = {HH, HT, TH, TT}
Probability of getting one head = 2/4 = 1/2.

2. Venn Diagrams (Two Sets Only)

• Venn diagrams show relationships between two sets (e.g., A and B).
• Key regions:
  • A only: In A but not B.
  • B only: In B but not A.
  • A ∩ B (A and B): In both A and B.
  • Outside A and B: In neither A nor B.

Rules:

• P(A or B) = P(A) + P(B) – P(A and B)
• P(A and B) = Probability of both events happening.

3. Tree Diagrams (With Replacement)

• Used for two or more events occurring one after another.
• With replacement = probabilities stay the same after each event.
• Probabilities are shown along the branches, outcomes at the end.

Multiplying Rule:

• To find the probability of a specific combined outcome:

Multiply probabilities along the branches.text{Multiply probabilities along the branches.}

Adding Rule:

• To find the probability of one of several paths:

Add probabilities of the different paths.text{Add probabilities of the different paths.}

Example:

• Toss a coin twice:
  • Probability of HH = (1/2) × (1/2) = 1/4.
  • Probability of HT or TH = (1/2 × 1/2) + (1/2 × 1/2) = 1/2.

4. Important Tips

• Sample space diagrams: list clearly without missing outcomes.
• Venn diagrams: always subtract overlap when needed.
• Tree diagrams:
  • Multiply along branches.
  • Add separate branches if asked for “either” outcomes.
• Probabilities must always add up to 1 at each stage.

This cheat sheet fully covers all CORE-level skills for calculating probabilities using sample spaces, Venn diagrams (two sets only), and tree diagrams (with replacement).