Venn diagrams and probability
Maths · Key Stage 3 · The School
Sets and circles
A VENN DIAGRAM shows sets as circles inside a rectangle representing everything possible. Where two circles overlap shows items belonging to BOTH sets. Outside both circles, but inside the rectangle, shows items belonging to NEITHER.
Reading AND, OR, NOT
P(A AND B) is the overlap region alone. P(A OR B) is everything in either circle, counting the overlap only once. P(NOT A) is everything outside circle A, including the parts of circle B that do not overlap with A. Naming which region a question wants, before counting anything, prevents nearly every error here.
Building one from scratch
Of 30 students, 18 study French, 12 study Spanish, and 5 study both. To draw the diagram: the overlap is 5. French-only is 18 − 5 = 13. Spanish-only is 12 − 5 = 7. Neither is 30 − 13 − 5 − 7 = 5. Check: 13 + 5 + 7 + 5 = 30, which must match the total or a region is wrong.
Using the French/Spanish diagram: P(studies French) = 18/30 (the whole French circle, overlap included). P(studies both) = 5/30 (the overlap alone). P(studies French OR Spanish) = (13 + 5 + 7)/30 = 25/30, NOT 18/30 + 12/30 = 30/30 — adding the two circle totals directly double-counts the 5 students in the overlap. P(studies neither) = 5/30. Every one of these reads straight off the diagram once it is correctly filled in, which is the entire point of drawing it before calculating anything.