下图显示了一个半径为 40 cm 的圆形靶盘，由两个半径分别为 12 cm 和 24 cm 的同心圆分成三个区域：金色、红色和蓝色。 向靶盘射箭，射中金色、红色和蓝色区域的得分分别为 10 分、6 分和 3 分。落在靶盘外的箭得 0 分。 亚历克斯向靶盘射了一支箭。他的箭落在靶盘上的概率为 $p$，其中 $p > 0.5$。如果他的箭落在靶盘上，则箭落在靶盘上任何位置的可能性是相等的。设 $X$ 表示单次射击所得的分数。 你可以假设箭不会落在任何区域的边界上。

1. Calculate the total area of the target board and the area of the Red region.

Area of target board $= \pi(40)^2 = 1600\pi\text{ cm}^2$

Area of Red region $= \pi(24)^2 - \pi(12)^2 = 576\pi - 144\pi = 432\pi\text{ cm}^2$

2. Determine $\mathrm{P}(X=6)$ by multiplying the probability of landing on the board by the conditional probability of landing in the Red region.

$\mathrm{P}(\text{Red} \mid \text{lands on board}) = \frac{432\pi}{1600\pi} = \frac{27}{100} = 0.27$

$\mathrm{P}(X=6) = \mathrm{P}(\text{lands on board}) \times \mathrm{P}(\text{Red} \mid \text{lands on board})$
$\mathrm{P}(X=6) = p \times 0.27 = 0.27p$

1. Determine the full probability distribution of $X$.

$\mathrm{P}(X=10) = \frac{\pi(12)^2}{\pi(40)^2} p = \frac{144\pi}{1600\pi} p = 0.09p$
From part (a), $\mathrm{P}(X=6) = 0.27p$.
$\mathrm{P}(X=3) = \frac{\pi(40)^2 - \pi(24)^2}{\pi(40)^2} p = \frac{1600\pi - 576\pi}{1600\pi} p = 0.64p$
$\mathrm{P}(X=0) = 1 - p$

2. Calculate the expected value, $\operatorname{E}(X)$.

$\operatorname{E}(X) = 10(0.09p) + 6(0.27p) + 3(0.64p) + 0(1-p)$
$\operatorname{E}(X) = 0.9p + 1.62p + 1.92p = 4.44p$

3. Calculate the expected value of the squares, $\operatorname{E}(X^2)$.

$\operatorname{E}(X^2) = 10^2(0.09p) + 6^2(0.27p) + 3^2(0.64p) + 0^2(1-p)$
$\operatorname{E}(X^2) = 100(0.09p) + 36(0.27p) + 9(0.64p)$
$\operatorname{E}(X^2) = 9p + 9.72p + 5.76p = 24.48p$

4. Form the variance equation and solve for $p$.

$\operatorname{Var}(X) = \operatorname{E}(X^2) - [\operatorname{E}(X)]^2$
$5.464476 = 24.48p - (4.44p)^2$
$5.464476 = 24.48p - 19.7136p^2$
$19.7136p^2 - 24.48p + 5.464476 = 0$
Using the quadratic formula:
$p = \frac{-(-24.48) \pm \sqrt{(-24.48)^2 - 4(19.7136)(5.464476)}}{2(19.7136)}$
$p = \frac{24.48 \pm \sqrt{168.3724237}}{39.4272}$
$p = 0.95 \quad \text{or} \quad p \approx 0.2918$
Since it is given that $p > 0.5$, we reject $p \approx 0.2918$.
$p = 0.95$