7 A clinic deals only with flu vaccinations. The number of patients arriving every 15 minutes is modelled by the random variable \(X\) with distribution \(\operatorname { Po } ( 4.2 )\).
- State two assumptions required for the Poisson model to be valid.
- Find the probability that
(a) at least 1 patient will arrive in a 15-minute period,
(b) fewer than 4 patients will arrive in a 10-minute period. - The clinic is open for 20 hours each week. At the beginning of one week the clinic has enough vaccine for 370 patients. Use a suitable approximation to find the probability that this will not be enough vaccine for that week.