2011 YJC P2 Q8

A fruit seller grades apples according to their mass. It is given that the mass of a randomly chosen apple follows a normal distribution with mean \(\mu \)g and standard deviation \(30\)g. Apples with a mass exceeding \(150\)g are graded as ‘large’ while apples with a mass less than \(70\)g are graded as ‘small’. The proportion of ‘large’ apples is the same as the proportion of ‘small’ apples.

1. Explain why \(\mu \) is \(110\)g.[1]
2. Find the probability that the total mass of two randomly chosen apples exceeds \(230\)g.[2]
3. Find the probability that a buyer who picks \(50\) apples randomly will have at least three apples which are graded as ‘large’.[4]

The fruit seller also grades oranges according to their mass. It is given that the mass of a randomly chosen orange has an independent normal distribution with mean \(190\)g and standard deviation \(24\)g. The fruit seller sells the apples at \(\$0.20\) per \(100\)g and the oranges at \(\$1.50\) per \(100\)g.

4. Find the probability that the average cost of an apple and two oranges exceeds \(\$0.25\).[3]