2018 DHS P2 Q8

An interactive simulation ride allows a group of \(5\) riders to take the ride at a time. The ride time, \(X\) minutes, follows a normal distribution with mean \(\mu \) minutes and standard deviation \(2\) minutes. The ride starts promptly at 10 am daily with no wait time between any groups of \(5\) riders. There are only \(4\) scheduled rides every morning. At 10 am on a particular morning, there are already \(20\) people queuing for the ride. It is assumed that all the people in the queue will take the ride based on the sequence of the queue and the ride times are independent.

1. Show that \(\mu =14\), correct to the nearest integer, if \(\mathrm{P}\left( \mu < X < 16 \right)=0.35\).[2]

For the rest of the question, use \(\mu =14\) for your calculations. A ride is considered long if it has a ride time of at least \(15\) minutes.

2. Find the probability that the \({{12}^{\mathrm{th}}}\) person in the queue took the ride before 10.30 am on that morning.[3]
3. Show that the probability of having at least \(2\) long rides on that morning is \(0.363\).[2]
4. Given that there are at least \(2\) long rides on that morning, find the probability that none of these long rides are consecutive.[2]