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

(i)
Let X and Y be the number of originals and prints sold in a week, respectively.
$X \sim \mathrm{Po}(2)$

$Y \sim \mathrm{Po}(11)$

(a)$\mathrm{P}(Y > 8) = 1 - \mathrm{P}(Y \le 8) = 0.768$

(b) $X + Y \sim \mathrm{Po}(13)$

$\mathrm{P}(X + Y < 15) = 1 - \mathrm{P}(X + Y \le 14) = 0.675$

(ii) Let W be the number of originals sold in $n$ weeks.

$W \sim \mathrm{Po}(2n)$

$\mathrm{P}(W < 3)<0.01$

$\mathrm{P}(W = 2) + \mathrm{P}(W = 1) + \mathrm{P}(W = 0)<0.01$

$e^{-2n} + e^{-2n}(2n) + e^{-2n}\frac{(2n)^{2}}{2}<0.01$

$e^{-2n}(1 + 2n + 2n^{2})<0.01$

Using GC, smallest possible n = 5.

(iii) Since $n=52$ is large, by Central Limit Theorem,

$B_1 + ... + B_52 \sim N(572, 572)$ approximately.

$\mathrm{P}(B_1 + ... + B_52 > 550) = 0.821$

(iv) Firstly the mean number of paintings sold per week may not be constant due to seasonal factors. Secondly, the sales might not be independent as a buyer might have preferential taste for particular artists.