下图显示了一个半径为 40 cm 的圆形靶盘，它被半径分别为 12 cm 和 24 cm 的两个同心圆划分为三个区域：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（Alex）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，则箭落在靶盘上任意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 area of the Red region.

The Red region is the annulus between the circles of radii 12 cm and 24 cm.
$\text{Area}_{\text{Red}} = \pi(24^2) - \pi(12^2) = 576\pi - 144\pi = 432\pi\text{ cm}^2$

2. Calculate the total area of the target board.

The target board has a radius of 40 cm.
$\text{Area}_{\text{Total}} = \pi(40^2) = 1600\pi\text{ cm}^2$

3. Calculate the probability of scoring 6 points.

Since the arrow is equally likely to hit any position on the board if it lands on it, the conditional probability of hitting the Red region is the ratio of its area to the total area. We multiply this by the probability $p$ that the arrow lands on the board.
$P(X=6) = p \times \frac{\text{Area}_{\text{Red}}}{\text{Area}_{\text{Total}}} = p \times \frac{432\pi}{1600\pi} = p \times 0.27 = 0.27p$

1. Find the probability distribution of $X$ by determining the probabilities of scoring 10, 3, and 0 points in terms of $p$.

For a single shot, the arrow lands on the board with probability $p$. Given that it lands on the board, the probability of hitting a specific region is proportional to its area.
Total area of the board = $\pi(40)^2 = 1600\pi$
Area of Gold region = $\pi(12)^2 = 144\pi$
Area of Blue region = $\pi(40)^2 - \pi(24)^2 = 1600\pi - 576\pi = 1024\pi$

$P(X=10) = p \times \frac{144\pi}{1600\pi} = 0.09p$
$P(X=3) = p \times \frac{1024\pi}{1600\pi} = 0.64p$

From part (a), $P(X=6) = 0.27p$.
The probability of missing the board entirely (scoring 0) is:
$P(X=0) = 1 - p$

2. Express the expected values $E(X)$ and $E(X^2)$ in terms of $p$.

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

$E(X^2) = 10^2(0.09p) + 6^2(0.27p) + 3^2(0.64p) + 0^2(1-p)$
$E(X^2) = 9p + 9.72p + 5.76p = 24.48p$

3. Set up a quadratic equation in $p$ using the given variance.

Using the variance formula $\operatorname{Var}(X) = E(X^2) - [E(X)]^2$:
$5.464476 = 24.48p - (4.44p)^2$
$5.464476 = 24.48p - 19.7136p^2$
$19.7136p^2 - 24.48p + 5.464476 = 0$

4. Solve the quadratic equation and select the valid root.

Using a graphing calculator to solve the quadratic equation:
$p = 0.95 \quad \text{or} \quad p \approx 0.29178$

Since it is given in the question that $p > 0.5$, we reject $p \approx 0.29178$.
$p = 0.95$