2025 YIJC P2 Q9

In this question, you should state the parameters of any normal distributions you use.

A confectionary bakes egg tarts for sale. The masses of these egg tarts have a normal distribution and \(26\)% of them have a mass greater than \(65\) grams. An egg tart is equally likely to have a mass less than \(50\) grams as it is to have a mass greater than \(70\) grams.

1. Find the mean and variance of this distribution.[3]

The confectionary also bakes donuts and buns. The masses in grams of the donuts have the distribution \(\mathrm{N}(80, 10^2)\) and the masses in grams of the buns have the distribution \(\mathrm{N}(70, 6^2)\).

2. Find the probability that the mass of a randomly chosen donut is within \(5\) grams from its mean.[1]
3. Find the probability that the mass of a randomly chosen bun is less than the mass of a randomly chosen donut.[2]

The confectionary also tops its donuts and buns with chocolate. A chocolate topping increases the mass of a donut by \(10\)% and the mass of a bun by \(20\)%.

4. Find the probability that the total mass, with chocolate topping, of \(3\) randomly chosen donuts and \(5\) randomly chosen buns exceeds \(700\) grams.[4]