2019 SAJC H1 Promo Q5

An athlete who is training for a marathon aims to complete multiple laps on a running track while running at a constant speed. In practice, the athlete’s judgement of pace is not accurate, and the time taken to complete a randomly chosen lap, \(X\), may be taken to be normally distributed with mean \(\mu \) seconds and standard deviation \(\sigma \) seconds. The time taken for a lap may be assumed to be independent of the time taken for another lap.

1. Given that \({\mathrm{P}\left( X < 64 \right)=\mathrm{P}\left( X > 66 \right)=0.2}\), find the value of \(\mu \) and show that the value of \(\sigma \) is \(1.19\).[3]
2. Find the probability that each of three randomly chosen laps will take less than \(66\)s.[2]
3. Find the probability that at least \(5\) out of the \(40\) randomly chosen laps will take more than \(66\)s.[3]

As part of the endurance training, the athlete aims to complete many uphill runs at a constant speed. The time taken for an uphill run may be assumed to be independent of the time taken for another uphill run. It is known that the time taken by the athlete to complete one uphill run is \(1.8\) times the time taken to complete one complete lap on the running track.

4. Find the probability that the time taken to complete two randomly chosen uphill runs exceeds twice the time taken to complete one randomly chosen lap of a running track by more than \(110\)s.[5]