2025 NJC P2 Q6

A circular board has \(6\) sectors labelled \(A\), \(B\), \(C\), \(D\), \(E\) and \(F\). A game is played by placing a pawn on sector \(A\). It is then moved clockwise around the board according to the number shown on the top face of a fair die when it is tossed. The die has faces labelled \(2\), \(2\), \(3\), \(4\), \(4\), \(5\). For example, an initial toss of \(4\) would move the pawn from sector \(A\) to sector \(E\). The player continues tossing the die and moving the pawn accordingly. When the pawn lands on sector \(F\), the game ends. Let \(T\) be the number of tosses required to move the pawn until the game ends.

1. Show that \(\mathrm{P}(T=3)=\frac{5}{36}\). Hence, copy and complete the probability distribution table for \(T\).[4]

   \(t\)	\(1 \)	\(2 \)	\(3 \)	\(4 \) or more
   \( \mathrm{P} (T=t) \)

The score \(S\) for each game is given by \(S=5T\), for \(t=1,2,3\) and \(s=0\) for \(t\ge 4\).

2. Find \(\mathrm{E}(S)\) and \(\mathrm{Var}(S)\).[3]