1 A study undertaken by Goodhealth Hospital found that the number of patients each month, $X$, contracting a particular superbug can be modelled by a Poisson distribution with a mean of 1.5 .\\
(a) (i) Calculate $\mathrm { P } ( X = 2 )$.\\
(ii) Hence determine the probability that exactly 2 patients will contract this superbug in each of three consecutive months.\\
(b) (i) Write down the distribution of $Y$, the number of patients contracting this superbug in a given 6-month period.\\
(ii) Find the probability that at least 12 patients will contract this superbug during a given 6-month period.\\
(c) State two assumptions implied by the use of a Poisson model for the number of patients contracting this superbug.

\hfill \mbox{\textit{AQA S2  Q1}}