2020 NYJC P2 Q7

A jar contains five \(20\)-cent coins and \(n\) \(50\)-cent coins, where \(n\ge 2\). In a game, Lee removes coins at random from the jar, one at a time, until he has at least \(70\) cents. He scores zero points if he removes only two coins, otherwise his score is half the number of coins he removed. The score Lee obtains is denoted by the random variable \(S\).

1. Show that \(\mathrm{P}\left( S=0 \right)=\frac{n\left( n+9 \right)}{\left( n+5 \right)\left( n+4 \right)}\).[1]
2. Find \(\mathrm{P}\left( S=s \right)\) for all other possible values of \(s\).[2]
3. Show that \(\mathrm{E}\left( S \right)=\frac{30}{\left( n+5 \right)\left( n+3 \right)}\) and \(\mathrm{Var}\left( S \right)=\frac{15n\left( 3{{n}^{2}}+40n+113 \right)}{{{\left( n+5 \right)}^{2}}\left( n+4 \right){{\left( n+3 \right)}^{2}}}\).[5]