N2015 P2 Q12

In this question you should state clearly the values of the parameters of any normal distribution you use.

The masses in grams of apples have the distribution \(\mathrm N(300,20^2)\) and the masses in grams of pears have the distribution \(\mathrm N(200,15^2)\). A certain recipe requires \(5\) apples and \(8\) pears.

1. Find the probability that the total mass of \(5\) randomly chosen apples is more than \(1600\) grams.[2]
2. Find the probability that the total mass of \(5\) randomly chosen apples is more than the total mass of \(8\) randomly chosen pears.[3]

The recipe requires the apples and pears to be prepared by peeling them and removing the cores. This process reduces the mass of each apple by \(15\%\) and the mass of each pear by \(10\%\).

3. Find the probability that the total mass, after preparation, of \(5\) randomly chosen apples and \(8\) randomly chosen pears is less than \(2750\) grams.[4]