N2014 P2 Q11

An art dealer sells original paintings and prints. The numbers of original paintings and prints sold in a week have independent Poisson distributions with means \(2\) and \(11\) respectively.

1. Find the probability that, in a randomly chosen week,
   1. more than \(8\) prints are sold,[2]
   2. fewer than \(15\) original paintings and prints in total are sold.[2]
2. The probability that fewer than \(3\) original paintings are sold in \(n\) weeks is less than \(0.01\). Write down an inequality in terms of \(n\) and hence find the least possible value of \(n\).[5]
3. Use a suitable approximation to find the probability that more than \(550\) prints are sold in \(52\) weeks.[3]
4. Give two reasons why the assumptions made in using these distributions may not be valid in this context.[2]