A hospital specializes in treating overweight patients. These patients have weights that are independently, normally distributed with mean 200 kg and standard deviation 15 kg. The elevator in the hospital will break if the total weight of people inside it exceeds 1150 kg. Six patients enter the elevator.

Find the probability that the elevator breaks.

let \(W = \sum\limits_{i = 1}^6 {{w_i}} \)     (M1)

\({w_i}{\text{ is N}}(200,{\text{ 1}}{{\text{5}}^2})\)

\({\text{E}}(W) = \sum\limits_{i = 1}^6 {{\text{E}}({w_i}) = 6 \times 200 = 1200} \)     A1

\({\text{Var}}(W) = \sum\limits_{i = 1}^6 {{\text{Var}}({w_i}) = 6 \times {{15}^2} = 1350} \)     A2

\(W{\text{ is N}}(1200,{\text{ 1350}})\)     (M1)

\({\text{P}}(W > 1150) = 0.913\) by GDC     A1A1

Note: Using 6 times the mean or a lower bound for the mean are acceptable methods.

[7 marks]

Candidates will often be asked to solve these problems that test if they can distinguish between a number of individuals and a number of copies. The wording of the question was designed to make the difference clear. If candidates wrote \({w_1} +  \ldots  + {w_6}\) in (a) and 12w in (b), they usually went on to gain full marks.