The diagram below shows a circular target board of radius $40$ cm, divided by two concentric circl

Question

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

Accepted Answer

p=0.95

2025 VJC P2 Q8

2025 VJC P2 Q8