2018 TGM P2 Q6

Coloured pens are packed into boxes of \(30\) pieces by a manufacturer. Each box is made up of randomly chosen coloured pens. The manufacturer produces a large number of pens every day. On average, \(65\%\) of the pens are blue. Explain why binomial distribution is appropriate for modelling the number of blue pens in a box.[2]

1. Find the probability that a randomly chosen box of pens contains at least \(19\) blue pens.[2]
2. A customer buys \(50\) randomly chosen boxes of pens.
   1. Find the probability that no more than \(29\) of these boxes contain at least \(19\) blue pens.[2]
   2. Using a suitable approximation, find the probability that the mean number of blue pens in a box is more than \(20\).[2]

It is given that the \(p\%\) of the pens are red. The probability that there is at most \(7\) red pens in a box, is \(0.76079\), correct to \(5\) significant figures. Hence find the value of \(p\).[3]