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 $\text{P}\left( X<175 \right)=\text{P}\left( Y>150 \right)$, show that ${{\mu }_{1}}+{{\mu }_{2}}=325$.
[2]
For the rest of this question, assume that ${{\mu }_{1}}=180$.
(i)
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]
(ii)
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.
(iii)
State, in terms of $p$, the probability that exactly one of the three customers receives the drink free of charge.
[1]
(iv)
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]
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