2025 VJC P2 Q8

The diagram below shows a circular target board of radius \(40\) cm, divided by two concentric circles of radii \(12\) cm and \(24\) cm, into three regions, Gold, Red and Blue.

Arrows are shot at the target board and the scores obtained for hitting the Gold, Red and Blue regions are \(10\) points, \(6\) points and \(3\) points respectively. Arrows that land outside the target board score \(0\) points.

Alex shoots one arrow at the target board. The probability that his shot lands on the target board is \(p\), where \(p > 0.5\). If his shot lands on the target board, the arrow is equally likely to hit any position on the board. Let \(X\) represent the score obtained from a single shot.

You may assume that the arrow does not land on the boundary of any region.

1. Show that the \(\mathrm{P}(X=6) = 0.27p\).[1]
2. Given that \(\mathrm{Var}(X) = 5.464476\), find the value of \(p\).[4]