All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
Firstly, gold coins are scattered randomly in a randomly chosen region of $1m^2$.
Secondly, the mean number of gold coins found in any randomly chosen region of $1m^2$ is constant.
Lastly, the finding of a gold coin in a randomly chosen region of area $1m^2$ is independent of the finding of another gold coins.

(ii)
Let X denote the number of gold coins in a region of $1m^2$.
$\mathrm{P}(X \ge 3)$

$= 1 - \mathrm{P}(X \le 2)$

$= 0.0474$

(iii)
Let Y denote the number of gold coins found in a region of $x~m^2$.
$Y ~\sim~ \mathrm{Po} (0.8x)$

$\mathrm{P} (Y = 1) = 0.2$

Using the Graphing Calculator, $x = 0.324$

(iv)
Let W denote the number of gold coins found in a region of $100m^2$
$W ~\sim~ \mathrm{Po} (80)$

Since $\lambda = 80 > 10$, $W ~\sim~ \mathrm{N} (80, 80)$ approximately

$\mathrm{P} (W \ge 90)$
$= \mathrm{P} (W \ge 89.5)$ by continuity correction
$\approx 0.144$

(v)
Let C and S denote the number of gold coins and pottery shards in in a region of $50m^2$ respectively.

$C ~\sim~ \mathrm{Po} (40)$
Since $\lambda = 40 > 10$, $C ~\sim~ \mathrm{N} (40, 40)$ approximately.
$S ~\sim~ \mathrm{Po} (150)$
Since $\lambda = 150 > 10$, $S ~\sim~ \mathrm{N} (150, 150)$ approximately.

$C + S ~\sim~ \mathrm{N} (190, 190)$ approximately

$\mathrm{P} (C + S \ge 200)$
$\mathrm{P} (C + S \ge 199.5)$
$\approx 0.245$

(vi)
$S + 3C ~\sim~ \mathrm{N} (30, 510)$ approximately

$\mathrm{P} (S + 3C \ge 0)$
$\mathrm{P} (S + 3C \ge -.05)$ by continuity correction
$\approx 0.912$

For (iii), students can also introduce the poisson formula from MF15 and solve by hand. Please rm. to apply continuity correction for (iv). I took awhile to understand (vi) too. :/ Students should read carefully if their english is as bad as mine. For (vi), students should note that it is necessary that they approximate to normal before consider the difference of the distributions. Be reminded that we cannot take the difference of poisson distributions.

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