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

OCR references P8 and P9 cover tree diagrams for combined events: independent events, and at Higher tier conditional events where items are taken without replacement. A tree diagram organises a sequence of events so that probabilities can be multiplied along branches and added across outcomes. Tree diagrams are a high-value topic on every tier, and the without-replacement case is a reliable source of the harder Higher marks.

Building a tree diagram

A tree diagram shows a sequence of events stage by stage.

So for two flips of a biased coin with $P(\text{head}) = 0.6$, the first stage has branches H ($0.6$) and T ($0.4$), and each splits again into H and T. The four end paths are HH, HT, TH and TT. Labelling each branch with its probability before doing any arithmetic keeps the diagram reliable.

Multiplying and adding

Two rules turn a tree diagram into a probability.

So the probability of two heads is $0.6 \times 0.6 = 0.36$, and the probability of exactly one head is $P(\text{HT}) + P(\text{TH}) = 0.6 \times 0.4 + 0.4 \times 0.6 = 0.48$. The phrase "at least one" is usually fastest via the complement: $P(\text{at least one head}) = 1 - P(\text{no heads})$. For two flips that is $1 - 0.4 \times 0.4 = 1 - 0.16 = 0.84$, which is far quicker than adding the three separate "one or two heads" paths.

A useful check is that the probabilities of all the end paths must sum to $1$. For the two-flip example, $0.36 + 0.24 + 0.24 + 0.16 = 1$, confirming the branch probabilities are consistent. If your end paths do not sum to $1$, there is an arithmetic error somewhere on the tree, so this check is worth doing before reading off the final answer.

Independent versus without replacement

Whether the branch probabilities change is the key distinction.

For independent events (such as repeated coin flips, or picking with replacement), the second-stage probabilities equal the first-stage ones, because the situation resets. For events without replacement, the first pick is not returned, so the totals change: picking two counters from $10$ leaves only $9$ for the second pick, and the favourable count drops too. Recognising which case applies is the decisive reasoning step.

Why tree diagrams matter

Tree diagrams handle any sequence of events and are the natural tool for "two picks" or "two days" questions, which OCR sets frequently. They make the AND and OR rules visual, and the without-replacement case tests conditional reasoning that anticipates A-level probability. Showing the diagram with correct branch probabilities secures method marks even if the final arithmetic slips, and OCR rewards the explicit "exactly one" or "at least one" reasoning.