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

(i)
(a)
$\mathrm{P} (test ~postiive)$

$= 0.001 \times 0.995 + 0.999 \times 0.005$

$= 0.00599$

(b)
$\mathrm{P} (disease~ | ~test positive)$

$\frac{\mathrm{P} (disease ~\cap~ test positive)}{\mathrm{P}(test ~positive)}$

$\frac{0.001 \times 0.995}{0.00599}$

$\approx 0.166$

(ii)
$\mathrm{P}(disease~ |~ test positive ) = 0. 75$

$\frac{\mathrm{P} (disease ~\cap~ test positive)}{\mathrm{P}(test~ positive)} = 0.75$

$\frac{0.001p}{0.001p + 0.999(1-p)} = 0.75$

Using the Graphing Calculator, $p \approx 0.999666$