Probability: combined events and tree diagrams

Basic probability

P(event) = number of favourable outcomes ÷ total number of equally likely outcomes.

P(A does not happen) = 1 − PA.

Independent events

Two events are independent if the outcome of one does not affect the probability of the other.

P(A and B) = PA × PB.

Example: Rolling a die twice. P(6 then 6) = 1/6 × 1/6 = 1/36.

Dependent events (conditional probability)

When items are drawn without replacement, probabilities change after each draw.

P(A then B without replacement): after drawing A, the denominator decreases by 1.

Tree diagrams

Each branch is labelled with a probability. Probabilities along each path are multiplied (AND). Probabilities from different paths are added (OR).

Structure rules:

• Branches from any one node must sum to 1.
• For "at least one" problems: use P(at least one) = 1 − P(none).

Edexcel exam style

Edexcel Papers 2/3 frequently test:

1. Tree diagrams with "without replacement" (higher demand — conditional probabilities differ on second set of branches).
2. "Show that" probability: set up the tree, then verify a given total probability.
3. Venn diagrams for mutually exclusive or overlapping events.