2023 JPJC P2 Q11

A fair cubical die has faces each labelled with one of the four numbers: \(1\), \(2\), \(3\) and \(4\). The die is thrown once and the number on the uppermost face is the score \(T\).

It is given that the mode of \(T\) is \(2\) and that \({\mathrm{P}\left( T=1 \right)=\mathrm{P}\left( T=4 \right)=\frac{1}{6}}\). Determine the probability distribution of \(T\).[2]

In a game at a carnival, a player repeatedly throws the cubical die. If \(T\) is an odd number, the player loses and the game ends. Otherwise, the player continues to throw the die until a \(4\) is shown and the game ends. Let \(Y\) be the number of throws that it takes for the player to throw the first \(4\). If the player loses the game, then \(Y=0\).

1. Show that \(\mathrm{P}\left( Y=0 \right)=\frac{2}{3}\).[2]

It is given that for \(\left| x \right|<1\), \({{1}^{2}}+{{2}^{2}}x+{{3}^{2}}{{ x }^{2}}+{{4}^{2}}{{x}^{3}}+\ldots =\frac{1+x}{{{\left( 1-x \right)}^{3}}}\).

2. If \(\mathrm{E}\left( Y \right)=\frac{2}{3}\), find \(\mathrm{Var}\left( Y \right)\).[3]