N2026 P2 Q10

In this question you should assume that the distributions are all independent of each other.

Mrs Ng lives in Aytown. She often drives to the neighbouring towns of Beetown and Seatown.

1. The time taken, in minutes, for Mrs Ng to drive from Aytown to Beetown using her usual route has the distribution \(\mathrm{N}(22,3^2)\).
   1. Find the probability that Mrs Ng takes between \(22.5\) and \(24\) minutes for a randomly chosen journey from Aytown to Beetown.[1]
      Mrs Ng often visits the shops in Beetown. The time, in minutes, she spends at the shops has the distribution \(\mathrm{N}(75,12^2)\). The time taken for the journey from Beetown to Aytown also has the distribution \(\mathrm{N}(22,3^2)\).
   2. Find the probability that a randomly chosen trip from Aytown to the shops at Beetown and back to Aytown by Mrs Ng takes less than \(2\) hours.[3]
      The time taken, in minutes, for Mrs Ng to drive from Aytown to Beetown using an alternative route has the distribution \(\mathrm{N}(25,4^2)\).
   3. Find the probability that a randomly chosen journey from Aytown to Beetown using the alternative route takes less time than a randomly chosen journey using Mrs Ng's usual route.[3]
2. The time taken, in minutes, for the journey from Aytown to Seatown is also normally distributed. The probability that a randomly chosen journey takes less than \(55\) minutes is \(0.873\) and the probability that a randomly chosen journey takes more than \(49\) minutes is \(0.471\).
   Find the probability that a randomly chosen journey from Aytown to Seatown takes between \(45\) and \(50\) minutes.[4]