Draw and use tree diagrams to calculate probabilities of combined events, including independent events and conditional events without replacement (Higher tier).

What this dot point is asking

The Eduqas probability content asks you to draw and use tree diagrams to find probabilities of combined events, covering both independent events (where one does not affect the other) and, at Higher tier, conditional events without replacement (where the first outcome changes the second probability). Tree diagrams organise the possibilities so the two governing rules, multiply along a path and add between paths, can be applied cleanly. They are a dependable multi-mark question at both tiers, with the without-replacement case being a signature Higher-tier task.

Building a tree diagram

A tree diagram branches once for each event, with every branch labelled with its probability.

So for two spins of a spinner that is red with probability $0.7$, the first event has branches red ($0.7$) and not red ($0.3$), and from each of those the second event branches again into red ($0.7$) and not red ($0.3$). The four paths are the four combined outcomes.

The two rules: multiply along, add between

The whole method rests on two operations.

So the probability of "exactly one red" in two spins is found by adding the red-then-not path and the not-then-red path. The phrase "at least one" is usually quickest via the complement: P(at least one red) $= 1 -$ P(no reds), where P(no reds) is a single path.

Independent events

Events are independent when the outcome of one does not change the probability of the other, which is the case for repeated spins, separate coins, or drawing with replacement.

For independent events, the second set of branch probabilities is identical to the first. So rolling a $6$ twice on a fair die has probability $\dfrac{1}{6} \times \dfrac{1}{6} = \dfrac{1}{36}$, because the first roll has no effect on the second.

Conditional events without replacement (Higher)

When an item is removed and not replaced, the second event's probabilities change.

The crucial adjustment is the denominator: after one item is removed, the total drops by one, and the relevant numerator drops too if an item of that type was taken.

Why tree diagrams matter

Tree diagrams make multi-stage probability manageable and are the standard tool Eduqas expects for combined events. The without-replacement case is conceptually important because it is the GCSE introduction to conditional probability, which the Venn diagram work also touches. Setting out the tree clearly, with correct branch probabilities, is what earns the method marks even if the final arithmetic slips.