2022 HCI P2 Q11

A cafeteria installed a vending machine which dispenses two types of coffee into disposable cups as follows:

(I) Black coffee, \(X\) ml, normally distributed with mean \({{\mu }_{1}}\) ml and standard deviation \(11.83\)ml, or

(II) White coffee, by first releasing a quantity of black coffee, \(Y\) ml, normally distributed with mean
\({{\mu }_{2}}\) ml and standard deviation \(11.83\) ml and then adding milk, \(M\)ml, normally distributed with mean \(35\) ml and standard deviation \(5.92\) ml.

Given that \(\mathrm{P}\left( X<175 \right)=\mathrm{P}\left( Y>150 \right)\), show that \({{\mu }_{1}}+{{\mu }_{2}}=325\).[2]

For the rest of this question, assume that \({{\mu }_{1}}=180\).

1. Find the probability that the total volume of \(2\) randomly chosen cups of black coffee exceeds twice the volume of a randomly chosen cup of white coffee by less than \(15\) ml. State an assumption that is needed for your calculation to be valid.[4]
2. The black coffee is sold at \(\$4\) per cup while the white coffee is sold at \(\$5\) per cup. The cost of ingredients for the black coffee is \(1\) cent per ml and the cost of ingredients for the milk is \(2\) cents per ml. \(100\)cups of coffee are sold per day, \(n\) of which are black coffee and the rest are white coffee. Find the largest value of \(n\) such that the probability of the total profit earned per day exceeding \(\$230\) is at least \(0.8\).[4]

To boost the sales of coffee from the vending machine, if the volume of the coffee dispensed falls below a certain level, the customer receives the drink free of charge. It is given that \(p\%\) of the customers who selected black coffee received the drink free of charge.

From a large number of customers who selected black coffee, three customers are chosen at random.

3. State, in terms of \(p\), the probability that exactly one of the three customers receives the drink free of charge.[1]
4. Given that the probability of exactly one of the three customers receiving the drink free of charge is at most \(0.1\), find the range of values of \(p\).[2]