Tree diagrams — independent and conditional events

Tree diagrams are a systematic way to represent all possible outcomes of two or more events and calculate their probabilities.

Structure of a tree diagram

Each branch represents one possible outcome. The probability is written on the branch. The probabilities of all branches from a single point must sum to 1.

At the end of each sequence of branches, multiply the probabilities along the path to find the combined probability.

To find the probability of a particular outcome that can happen in more than one way, add the probabilities of the relevant paths.

The two rules: × and +

• To find "A AND B": multiply along the branch.
• To find "A OR B": add the relevant branch-end probabilities.

Independent events

Two events are independent if the outcome of one does not affect the probability of the other. In a tree diagram for independent events, the probabilities on the second set of branches are the same regardless of what happened first.

Example: A bag contains 3 red balls and 2 blue balls. A ball is drawn, replaced, then a second is drawn.

• P(red) = 3/5 on both draws (replacement → independent).

P(both red) = 3/5 × 3/5 = 9/25. P(exactly one red) = (3/5 × 2/5) + (2/5 × 3/5) = 6/25 + 6/25 = 12/25.

Conditional probability (without replacement)

If items are NOT replaced, the second draw depends on the first → probabilities change.

Example: 3 red and 2 blue balls, drawn without replacement.

• If first draw is red: P(red 2nd) = 2/4 = 1/2; P(blue 2nd) = 2/4 = 1/2.
• If first draw is blue: P(red 2nd) = 3/4; P(blue 2nd) = 1/4.

P(both same colour) = P(RR) + PBB = (3/5 × 2/4) + (2/5 × 1/4) = 6/20 + 2/20 = 8/20 = 2/5.

CCEA exam tips

• Always show the tree diagram clearly with probabilities on branches.
• Check that branches at each node sum to 1 (this helps catch arithmetic errors).
• For "at least one" questions: calculate 1 − P(none).
• Read carefully whether replacement is used — this determines whether events are independent.