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

(i)
Let X, Y be the output daily from mines A and B, respectively.
$X ~\sim~ \mathrm{N}(50, \sigma^{2})$

$\mathrm{P}(X>75) = 0.0189$

$latex \mathrm{P}(X<75) = 0.9811$ $latex \Rightarrow \mathrm{P}(Z< \frac{25}{\sigma}) = 0.9811$ $latex \frac{25}{\sigma} = 2.07701689$ $latex \sigma^{2} = 144.88 \approx 145$ (ii) $latex Y ~\sim~ \mathrm{N}(75, 64)$ $latex Y_1 + \ldots +Y_7 ~\sim~ \mathrm{N}(525,448)$ $latex \mathrm{P}(Y_1 + \ldots +Y_7 < 500) = 0.119$ (iii) $latex Y_1 + \ldots + Y_7 - 2(X_1 + \ldots +X_5) ~\sim~ \mathrm{N} (25, 3348)$ $latex \mathrm{P}[Y_1 + \ldots + Y_7 - 2(X_1 + \ldots +X_5) > 0] = 0.667$