2017 TPJC P2 Q8

A manufacturing plant processes raw material for a supplier. An order placed with the plant is considered to be a bulk order when a worker is expected to process more than \(300\) kg (kilograms) of raw material.
Albert uses a machine to process \(X\) kg of raw material and Bob uses a separate machine to process \(Y\) kg of raw material on a working day. \(X\) and \(Y\) are independent random variables with the distributions \(\mathrm{N}\left( 296,{{8}^{2}} \right)\) and \(\mathrm{N}\left( 290,{{12}^{2}} \right)\) respectively.

1. Find the probability that Albert processes more than \(300\) kg of raw material on a randomly selected working day.[2]
2. Find the probability that, over a period of \(15\) independent working days, there are exactly four working days on which Albert processes more than \(300\) kg of raw material. [2]
3. Find the probability that the total amount of raw material Bob processes over two working days exceeds twice the amount of raw material Albert processes on one working day. [4]

The plant receives a bulk order and Albert wants to have a probability of at least \(0.95\) of meeting the order.

4. This can be done by changing the value of \(\mu \), the mean amount of raw material Albert processes using the machine, but the standard deviation remains unchanged. Find the least value of \(\mu \).[3]