Draw and use probability tree diagrams for two or more events, multiplying along branches and adding across routes, including independent events and conditional events without replacement (Higher tier).

What this dot point is asking

WJEC asks you to draw and use probability tree diagrams for two or more events, multiplying probabilities along the branches and adding across the separate routes that satisfy the condition. At Foundation and basic Higher level the events are usually independent (with replacement), and at Higher tier the more demanding cases are conditional, where the probabilities change because an item is not replaced. The tree organises the outcomes so the "and" and "or" rules apply cleanly, and it is one of the most reliably examined probability skills on Unit 2.

Building a tree diagram

Each event becomes a set of branches.

A clear, fully labelled tree is itself worth marks, and it prevents the counting errors that catch out candidates who try to do the calculation in their head.

Multiplying and adding

The two rules turn the tree into a calculation.

To find the probability of a particular sequence (one event and then another), multiply the probabilities along its branches. To find the probability of an event that can happen in several ways (one route or another), find each route's probability by multiplying, then add the routes. So "multiply along, add across" is the whole method.

Independent events

When the first event does not affect the second, probabilities stay fixed.

Events are independent if the outcome of one does not change the probabilities of the next, which happens when an item is replaced or each trial is separate (two spins, two coin flips). The second set of branch probabilities is then identical to the first. For a spinner landing on red with probability $0.3$, two independent spins both red is $0.3 \times 0.3 = 0.09$.

A common Foundation question asks for "at least one" of an outcome, which is best answered using the complement: $P(\text{at least one}) = 1 - P(\text{none})$. So for the same spinner, the probability of at least one red in two spins is $1 - (0.7 \times 0.7) = 1 - 0.49 = 0.51$, which is faster than adding the three separate "exactly one" and "exactly two" routes.

Without replacement (Higher)

When an item is removed and not replaced, the probabilities change.

Both the numerator and the denominator drop by one for the colour drawn, so always update the second-draw probabilities.

Why this matters

Tree diagrams are a high-value Unit 2 topic that brings together the "and" (multiply) and "or" (add) rules in a single, visual method. The Foundation and basic cases reward a neat, correctly labelled tree, while the Higher without-replacement questions, where probabilities change, are a reliable discriminator worth several marks. The disciplines of reducing the denominator after each draw and of adding every route that meets the condition (such as the two orders that give "different colours") are what separate full marks from a partial attempt.