2017 AJC P2 Q8

A biscuit manufacturer produces both cream and chocolate biscuits. Biscuits are chosen randomly and packed into boxes of \(10\). The number of cream biscuits in a box is denoted by \(X\).

1. On average, the proportion of cream biscuits is \(p\). Given that \(\mathrm{P}\left( X=1\,\,\mathrm{or}\,\,2 \right)=0.15\), write down an equation for the value of \(p\). Hence find the value(s) of \(p\) numerically.[3]
2. It is given instead that the biscuit manufacturer produces \(3\) times as many cream biscuits as chocolate biscuits.
   1. Find the most likely value of \(X\).[2]
   2. A random sample of \(18\) boxes is taken. Find the probability that at least \(3\) but fewer than \(7\) boxes have equal numbers of cream and chocolate biscuits.[3]

A box of biscuits is sold at \(\$10\). The manufacturer gives a discount of \(\$2\) per box to its premium customers. The mean and variance of the number of boxes sold per day to each type of customers (assuming independence) are as follows:

Find the approximate probability that the total amount collected per month from the sales of biscuits is not less than \(\$255,000\), assuming that there are \(30\) days in a month.[4]