2025 EJC P2 Q10

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

A drinks stall specialises in fruit smoothies, which contain apples. The apples used by the stall have masses, in grams, that follow the distribution \(\mathrm{N}(100, 6^2)\).

1. Find the probability that the mass of a randomly chosen apple is no more than \(105\) grams.[1]

Before the apples can be blended, they need to be prepared by peeling them and removing their cores. This preparation process reduces the mass of each apple by \(30\%\).

2. Find the probability that the mass of a randomly chosen apple, after preparation, is more than \(78\) grams.[3]

The stall sells its Signature smoothie, which uses \(2\) prepared apples, \(1\) peeled banana and \(12\) blackcurrants; all of these are randomly chosen. The stall also adds a fixed \(50\) grams of milk before blending everything together. You should ignore any changes in mass from the blending process.

• The masses of peeled bananas, in grams, follow the distribution \(\mathrm{N}(130, 8^2)\).

• The masses of blackcurrants, in grams, follow the distribution \(\mathrm{N}(2.5, 0.6^2)\).

3. \(90\%\) of the Signature smoothies have a mass of at least \(m\) grams. Find the value of \(m\).[4]
4. A teacher purchases \(25\) Signature smoothies for her form class. Find the probability that at least \(20\) of the Signature smoothies have masses between \(330\) and \(360\) grams.[3]

The stall also makes blackcurrant smoothies using \(n\) blackcurrants and a fixed \(175\) grams of milk.

5. Find the smallest value of \(n\) such that the probability of the mass of a blackcurrant smoothie exceeding \(300\) grams is at least \(0.95\).[3]