At a particular game stall, players throw darts at a target board to earn points for redeeming prizes. The target board can be modelled by three concentric circles with radii one unit, three units and five units respectively. A player will get a score of 50, 25 or 0 points if the dart lands on the respective regions labelled 50, 25 or 0 as shown in the diagram above. The probability that a dart lands on any region labelled 50, 25 or 0 is proportional to the area of the region. It can be assumed that the borders between the regions have negligible thickness and the dart will only land on the target board, which has a total area of $25\pi$ units$^2$. In a game, a player throws two darts at the target board, one after the other. The first dart will be removed before the next dart is thrown. Let $S$ be the combined score that the player achieves from the two throws.

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 $\mathrm{Var}(S) = \mathrm{E}(S^2) - [\mathrm{E}(S)]^2$:
$\mathrm{Var}(S) = 800 - 20^2$
$\mathrm{Var}(S) = 800 - 400 = 400$