The Venn diagram shows the number of members of a sports club who play badminton \(B\) and who play squash \(S\). (a) Find \(n(B)\). [1] (b) Find \(n\big((B...

Reading a Venn diagram means recognising which regions each piece of set notation covers. Here the diagram's four regions are the badminton-only region, the overlap of the two circles, the squash-only region and the outside.

(a) \(n(B)\) is the whole badminton circle, both its own region and the overlap it shares with squash: \(n(B)=20\) [B1]

(b) \((B\cup S)'\) is the region outside both circles, the members who play neither game: \(9\) [B1]

(c) \(B'\cap S\) means not badminton and squash, so it is the squash-only region: \(11\) [B1]

Since the four regions total \(40\) members, the overlap holds \(6\) and the badminton-only region holds \(20-6=14\).

(d) Playing both games is the overlap: \(P=\frac{6}{40}=\frac{3}{20}\) [B1] oe, dividing numerator and denominator by \(2\).

(e) Exactly one game means the two "only" regions, deliberately excluding the overlap:

\(P=\frac{14+11}{40}\) [M1] \(=\frac{25}{40}=\frac{5}{8}\) [A1] oe

Note the difference between \(n(B)=20\), which includes the members who also play squash, and the badminton-only count of \(14\). The word "exactly" is the signal to exclude the overlap.