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
$\approx 0.912$