2020 ASRJC P2 Q7

The masses, in grams, of mangoes have the distribution \(\mathrm{N}\,(200,{{30}^{2}})\) and the \(2\) masses, in grams, of oranges have the distribution \(\mathrm{N}\,(\mu ,{{\sigma }^{2}})\).

1. Let \(A\) be the average mass of a mango and two oranges. Given that \(\mathrm{P}\,(A<162)=\mathrm{P}\,(A>206)=0.06\), find \(\mu \). Hence form an equation involving \(\sigma \) and solve for \(\sigma \).[5]
2. Find probability that the total mass of \(3\) mangoes differs from \(3\) times the mass of an orange by at most \(136\) grams.[3]
3. Mangoes are sold at \(\$26\) per kilogram. Find the probability that the total selling price of \(6\) mangoes exceeds \(\$29.50\).[2]