SPM.1.AHL.TZ0.10

pestleMathematicsAIHLPaper 1SPM· ahl-4-15-central-limit-theoremsource ↗

In a coffee shop, the time it takes to serve a customer can be modelled by a normal distribution with a mean of 1.5 minutes and a standard deviation of 0.4 minutes.

Two customers enter the shop together. They are served one at a time.

Find the probability that the total time taken to serve both customers will be less than 4 minutes.

Clearly state any assumptions you have made.

Markscheme / solution

let $T$ be the time to serve both customers and ${T_i}$ the time to serve the $i$ th customer

assuming independence of ${T_1}$ and ${T_2}$ R1

$T$ is normally distributed and $T = {T_1} + {T_2}$ (M1)

$E\left( T \right) = 1.5 + 1.5 = 3$ A1

${\text{Var}}\left( T \right) = {0.4^2} + {0.4^2} = 0.32$ M1A1

$P\left( {T < 4} \right) = 0.961$ A1

[6 marks]

Examiners’ report

[N/A]