Use Venn diagrams and set notation (union, intersection and complement) to represent and count outcomes and to calculate probabilities, including conditional probability (Higher tier).

What this dot point is asking

OCR references P6 and S5 cover Venn diagrams and set notation: union, intersection and complement, used to organise data and calculate probabilities, including conditional probability at Higher tier. A Venn diagram sorts items into overlapping groups so that counts and probabilities can be read directly. This topic is assessed on every tier and connects probability to data handling, with the conditional-probability case being a demanding Higher skill.

Set notation

A few symbols describe how groups relate.

So if $A = \{1, 2, 3, 4\}$ and $B = \{3, 4, 5\}$, then $A \cap B = \{3, 4\}$, $A \cup B = \{1, 2, 3, 4, 5\}$, and $n(A \cup B) = 5$. Reading the symbols correctly is essential, because $\cap$ (intersection, "and") and $\cup$ (union, "or") are easily confused.

Filling a Venn diagram

The order of filling matters.

The reliable method is to work from the inside out: place the intersection (the overlap) first, then subtract it from each group total to find the "only" regions, then place anything outside all the circles. This avoids the most common error of double-counting the items that belong to both groups. The numbers in all regions should sum to the total being considered.

Reading probabilities from a Venn diagram

Once the diagram is filled, probabilities are simple counts.

So in a group of $40$ where $9$ are in the overlap of $A$ and $B$, $P(A \cap B) = \dfrac{9}{40}$. Worded conditions translate to regions: "neither" is outside both circles, "exactly one" is the two "only" regions combined, and "at least one" is the whole union.

The addition rule connects these regions: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$, where the overlap is subtracted because it would otherwise be counted twice. So if $20$ are in $A$, $15$ in $B$ and $8$ in both, then $n(A \cup B) = 20 + 15 - 8 = 27$. The same logic in probability gives $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, which is the probability "or" rule listed on the OCR formulae sheet. Recognising when to subtract the overlap is the key to avoiding the double-counting error.

Conditional probability (Higher)

Conditional probability narrows the total to a given group.

Why Venn diagrams matter

Venn diagrams organise overlapping categories cleanly, and OCR uses them for survey data, set listing, and probability. They make conditional probability concrete by showing exactly which total to use, anticipating A-level work. Filling the overlap first and reading the union, intersection and complement correctly secures the marks, and the set-notation symbols are tested directly. Translating worded conditions ("neither", "exactly one", "given that") into regions is the key AO2 skill.