There are 40 plot holders on a garden allotment site just outside the town. The site secretary is putting together a seed order for the spring, so she asks ...

With two overlapping sets the important point is that the 9 plot holders who grow both crops are already included in the 24 who grow tomatoes and in the 18 who grow onions. Subtracting the overlap from each total gives the numbers who grow only one crop, and every plot holder then sits in exactly one of four regions.

1. Complete the Venn diagram [2 marks]
   \[\text{tomatoes only} = 24 - 9 = 15, \qquad \text{onions only} = 18 - 9 = 9\] \[\text{neither} = 40 - (15 + 9 + 9) = 40 - 33 = 7\] The four regions add to \(15 + 9 + 9 + 7 = 40\), which is the number of plot holders, so the diagram is consistent.
2. Find \(n(T \cup O)\) [1 mark]
   The union is everyone inside at least one circle: \[15 + 9 + 9 = 33\] The formula gives the same result: \(n(T) + n(O) - n(T \cap O) = 24 + 18 - 9 = 33\). The overlap must be subtracted once, otherwise the 9 who grow both are counted twice.
3. Find \(n(T' \cap O')\) [1 mark]
   \(T' \cap O'\) is the set of plot holders who grow neither crop, the region outside both circles: \[40 - 33 = 7\]
4. Probability of tomatoes but not onions [2 marks]
   That is the tomatoes only region, 15 plot holders out of 40: \[P = \frac{15}{40} = \frac{3}{8}\] Using 24 instead of 15 would be answering "grows tomatoes", which includes the 9 who also grow onions and so is a different question.

Examination reminder: the phrase "but not" always points to a single region of the diagram, so fill in the diagram first and read the answer from it rather than working from the circle totals.