2023 ASRJC P2 Q9

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

The masses, in grams, of apples and pears have the independent distributions \(N\left( 260,\ {{15}^{2}} \right)\) and \(N\left( 210,\ {{10}^{2}} \right)\) respectively.

1. Find the probability that the mass of a randomly chosen apple is at least \(250\) grams.
2. Find the probability that the mass of a randomly chosen apple differs from the mass of a randomly chosen pear by more than \(55\) grams.[2]

A certain recipe requires \(5\) apples and \(6\) pears. The recipe requires the apples and pears to be prepared by removing the cores. This process reduces the mass of each apple by \(5\%\) and the mass of each pear by \(15\%\).

3. Find the probability that the total mass, after preparation, of \(5\) randomly chosen apples and \(6\) randomly chosen pears is at least \(2.3\) kilograms.[3]
4. Find \(\mathrm{P}\left( 3\left( 0.7A \right)-0.8\left( {{B}_{1}}+{{B}_{2}} \right)>200 \right)\). Explain, in the context of this question, what your answer possibly represents.[4]