Tree diagrams, Venn diagrams, sample space diagrams, independent and conditional probability, and set notation.

What this dot point is asking

AQA wants you to use sample space diagrams, tree diagrams and Venn diagrams to find probabilities of combined events, distinguish independent from conditional probability, and use basic set notation. These diagrams are the main tools for the longer probability questions on both papers.

Sample space diagrams

The strength of a sample space diagram is that, when every cell is equally likely, probability is just counting: the chance of a total of $7$ on two dice is the $6$ cells that sum to $7$ out of $36$, giving $\frac{6}{36} = \frac{1}{6}$. They become unwieldy for more than two stages, where a tree diagram is clearer.

Tree diagrams

Venn diagrams and set notation

Always fill the intersection first, then subtract to get the "only" regions, then place anything in neither set. The regions must total the whole group. The union $A \cup B$ counts everything in $A$ or $B$ (including the overlap once), which is why the general addition rule subtracts the intersection: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Conditional probability

To read a conditional probability from a Venn diagram, restrict attention to the "given" set: $P(\text{French} \mid \text{German})$ is the number who do both divided by the number who do German, not by the whole class. The key shift is the denominator: a conditional probability narrows the sample space to only those cases where the given event happened, so you divide by the size of that group rather than by the whole population.

Choosing the right diagram speeds you up. A tree diagram is best for a sequence of stages, especially "without replacement" problems where probabilities change at each step, because the branch structure makes the changing fractions explicit. A Venn diagram is best for overlapping categories, because it lays out the "both", "only", and "neither" regions you need for unions, intersections and conditional probabilities. A sample space diagram suits two simple events with equally likely outcomes, where counting cells is quickest.