5.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{78ba3acc-4cca-4d15-8362-a27e425c5859-16_517_881_210_593}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

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 $H \cap G$
\begin{enumerate}[label=(\alph*)]
\item Show that $\mathrm { P } ( F \cap H \cap G ) = \frac { 1 } { 7 } \mathrm { P } ( G )$
\item Prove that if two events $X$ and $Y$ are independent,then $X ^ { \prime }$ and $Y$ are also independent.
\item Hence find the value $k$ such that $\mathrm { P } \left( F ^ { \prime } \cap H \cap G \right) = k \mathrm { P } ( G )$
\item Show that $c = \frac { 4 } { 3 } a$

Given further that $\mathrm { P } ( F \mid H ) = \frac { 1 } { 5 }$
\item find an expression for $d$ in terms of $a$ ,and hence deduce the maximum possible value of $a$ .
\item Determine the possible values of $n$ .
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2023 Q5 [21]}}