5.
\begin{figure}[h]
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\caption{Figure 2}
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Figure 2 shows a partially completed Venn diagram of sports that a year group of students enjoy，where \(a , b , c , d\) and \(e\) are non－negative integers．
The diagram shows how many students enjoy a combination of football（ \(F\) ），golf（ \(G\) ） and hockey \(( H )\) or none of these sports．
There are \(n\) students in the year group．
It is known that
－ \(\mathrm { P } ( F ) = \frac { 3 } { 7 }\)
－ \(\mathrm { P } ( H \mid G ) = \frac { 1 } { 3 }\)
－\(F\) is independent of \(H \cap G\)
（a）Show that \(\mathrm { P } ( F \cap H \cap G ) = \frac { 1 } { 7 } \mathrm { P } ( G )\)
（b）Prove that if two events \(X\) and \(Y\) are independent，then \(X ^ { \prime }\) and \(Y\) are also independent．
（c）Hence find the value \(k\) such that \(\mathrm { P } \left( F ^ { \prime } \cap H \cap G \right) = k \mathrm { P } ( G )\)
（d）Show that \(c = \frac { 4 } { 3 } a\)
Given further that \(\mathrm { P } ( F \mid H ) = \frac { 1 } { 5 }\)
（e）find an expression for \(d\) in terms of \(a\) ，and hence deduce the maximum possible value of \(a\) ．
（f）Determine the possible values of \(n\) ．