在某个游戏摊位上，玩家向目标靶投掷飞镖以获得积分来兑换奖品。 目标靶可以建模为三个同心圆，其半径分别为一个单位、三个单位和五个单位。如图所示，如果飞镖落在标记为 50、25 或 0 的相应区域，玩家将分别获得 50 分、25 分或 0 分。飞镖落在标记为 50、25 或 0 的任何区域的概率与该区域的面积成正比。可以假设各区域之间的边界厚度忽略不计，且飞镖只会落在目标靶上，目标靶的总面积为 $25\pi$ 单位$^2$。 在一次游戏中，玩家向目标靶先后投掷两枚飞镖。在投掷下一枚飞镖之前，会先取下第一枚飞镖。设 $S$ 为玩家在两次投掷中获得的总得分。

1. Calculate the probability of scoring 50 and 25 on a single throw by finding the areas of the respective regions.

$\text{Area of region for } 50 = \pi(1)^2 = \pi$
$\text{Area of region for } 25 = \pi(3)^2 - \pi(1)^2 = 8\pi$
$\text{Total area} = 25\pi$
$P(50) = \frac{\pi}{25\pi} = \frac{1}{25}$
$P(25) = \frac{8\pi}{25\pi} = \frac{8}{25}$

2. Identify the score combinations that sum to 75 and compute the combined probability.

$\text{To obtain } S = 75\text{, the scores from the two independent throws must be 50 and 25.}$
$P(S = 75) = P(50 \text{ then } 25) + P(25 \text{ then } 50)$
$P(S = 75) = \left( \frac{1}{25} \times \frac{8}{25} \right) + \left( \frac{8}{25} \times \frac{1}{25} \right)$
$P(S = 75) = \frac{8}{625} + \frac{8}{625}$
$P(S = 75) = \frac{16}{625} \quad \text{(shown)}$

1. Determine the probability of getting each score for a single dart throw.

Let $X$ be the score from a single dart. The probability of landing on a region is proportional to its area. The total area of the board is $25\pi$.

$\mathrm{P}(X=50) = \frac{\pi(1)^2}{25\pi} = \frac{1}{25}$
$\mathrm{P}(X=25) = \frac{\pi(3)^2 - \pi(1)^2}{25\pi} = \frac{8\pi}{25\pi} = \frac{8}{25}$
$\mathrm{P}(X=0) = \frac{25\pi - \pi(3)^2}{25\pi} = \frac{16\pi}{25\pi} = \frac{16}{25}$

2. Identify the possible values of $S$ and calculate the probabilities for $S=0$ and $S=25$.

The possible combined scores for two independent throws are $0, 25, 50, 75$, and $100$.

$\mathrm{P}(S=0) = \mathrm{P}(X_1=0)\mathrm{P}(X_2=0) = \left(\frac{16}{25}\right)^2 = \frac{256}{625}$

$\mathrm{P}(S=25) = \mathrm{P}(X_1=0)\mathrm{P}(X_2=25) + \mathrm{P}(X_1=25)\mathrm{P}(X_2=0)$
$\mathrm{P}(S=25) = 2 \times \left(\frac{16}{25}\right)\left(\frac{8}{25}\right) = \frac{256}{625}$

3. Calculate the probabilities for $S=50$ and $S=100$.

$\mathrm{P}(S=50) = \mathrm{P}(0, 50) + \mathrm{P}(50, 0) + \mathrm{P}(25, 25)$
$\mathrm{P}(S=50) = 2 \times \left(\frac{16}{25}\right)\left(\frac{1}{25}\right) + \left(\frac{8}{25}\right)^2 = \frac{32}{625} + \frac{64}{625} = \frac{96}{625}$

$\mathrm{P}(S=100) = \mathrm{P}(X_1=50)\mathrm{P}(X_2=50) = \left(\frac{1}{25}\right)^2 = \frac{1}{625}$

4. Summarise the probability distribution of $S$ in a table, incorporating the result from part (a).

From part (a), $\mathrm{P}(S=75) = \frac{16}{625}$.
To check if the distribution is valid: $\frac{256 + 256 + 96 + 16 + 1}{625} = \frac{625}{625} = 1$.

The probability distribution of $S$ is:

1. State the probability distribution of $S$ from part (b) and calculate the expectation.

From part (b), the probability distribution of $S$ is:

• $\mathrm{P}(S=100) = \frac{1}{625}$
• $\mathrm{P}(S=75) = \frac{16}{625}$
• $\mathrm{P}(S=50) = \frac{96}{625}$
• $\mathrm{P}(S=25) = \frac{256}{625}$
• $\mathrm{P}(S=0) = \frac{256}{625}$

The expectation of $S$ is given by $\mathrm{E}(S) = \sum s \,\mathrm{P}(S = s)$:
$\mathrm{E}(S) = 100\left(\frac{1}{625}\right) + 75\left(\frac{16}{625}\right) + 50\left(\frac{96}{625}\right) + 25\left(\frac{256}{625}\right) + 0\left(\frac{256}{625}\right)$
$\mathrm{E}(S) = \frac{100 + 1200 + 4800 + 6400 + 0}{625}$
$\mathrm{E}(S) = \frac{12500}{625} = 20$

2. Calculate the expectation of $S^2$ and use it to find the variance.

Next, calculate $\mathrm{E}(S^2) = \sum s^2 \,\mathrm{P}(S = s)$:
$\mathrm{E}(S^2) = 100^2\left(\frac{1}{625}\right) + 75^2\left(\frac{16}{625}\right) + 50^2\left(\frac{96}{625}\right) + 25^2\left(\frac{256}{625}\right) + 0$
$\mathrm{E}(S^2) = \frac{10000 + 5625(16) + 2500(96) + 625(256)}{625}$
$\mathrm{E}(S^2) = \frac{10000 + 90000 + 240000 + 160000}{625}$
$\mathrm{E}(S^2) = \frac{500000}{625} = 800$

The variance of $S$ is given by $\mathrm{Var}(S) = \mathrm{E}(S^2) - [\mathrm{E}(S)]^2$:
$\mathrm{Var}(S) = 800 - 20^2$
$\mathrm{Var}(S) = 800 - 400 = 400$