Venn diagrams for two or three sets, set notation (union, intersection and complement), and using a completed Venn diagram to find probabilities including conditional probability (Higher tier).

What this dot point is asking

Edexcel expects you to use Venn diagrams to organise overlapping groups, to read and use set notation (union, intersection and complement), and to find probabilities from a completed Venn diagram, including conditional probability at Higher tier. Venn diagrams turn wordy "how many do both" problems into a clear picture, and set notation gives a precise language for the regions.

Set notation

Set notation names the regions of a Venn diagram precisely, and Edexcel uses these symbols in questions.

So $P(A \cap B)$ is the probability of being in the overlap, and $P(A \cup B)$ is the probability of being in either circle. Reading the symbols correctly is half the task.

Filling a Venn diagram

The reliable order is overlap first, then the parts of each set outside the overlap, then anything outside all sets.

For two sets, if $20$ like apples, $15$ like bananas, $8$ like both, then the overlap is $8$, apples-only is $20 - 8 = 12$, bananas-only is $15 - 8 = 7$. Starting with the overlap prevents double-counting, which is the single most common mistake.

Three-set Venn diagrams

With three sets there are more regions, but the same principle applies: work from the centre (in all three) outwards.

Probability and conditional probability

To find a probability from a Venn diagram, count the items in the region you want and divide by the total in the diagram. Conditional probability, written $P(A \mid B)$ ("A given B"), narrows the focus to set $B$ only: the denominator becomes the total in $B$, not the whole universal set. This is why, in the exam question, "coffee given tea" divides by the number of tea drinkers, not by everyone.

Reading probabilities of unions and intersections

Once a Venn diagram is filled, set-notation probabilities follow directly. $P(A \cap B)$ counts only the overlap; $P(A \cup B)$ counts every region inside either circle; and $P(A')$ counts everything outside $A$. A useful identity is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$, because adding the two circles double-counts the overlap, so you subtract it once. For example, if $P(A) = 0.5$, $P(B) = 0.4$ and $P(A \cap B) = 0.2$, then $P(A \cup B) = 0.5 + 0.4 - 0.2 = 0.7$. Translating the symbols into regions before counting prevents most errors.

Try this

Q1. In a group of $25$ people, $14$ play tennis, $11$ play golf and $5$ play both. How many play neither? [3 marks]

• Cue. Tennis only $9$, golf only $6$, both $5$, so $20$ play at least one; $25 - 20 = 5$ play neither.

Q2. From the same group, find $P(\text{plays golf} \mid \text{plays tennis})$. [2 marks]

• Cue. Tennis players total $14$; of these $5$ also play golf, so $\dfrac{5}{14}$.