CJC P2 Q9

A student reaches the bus-stop outside his house at \(7.20\) am every morning to take a bus to school and has to reach his school by \(7.40\) am. Assume that the waiting time for his bus is normally distributed with mean \(8\) minutes and variance \(5\) minutes\(^{2}\), and the journey time is normally distributed with mean \(11\) minutes and variance \(4\) minutes\(^{2}\).

1. Find the probability that he will take more than \(20\) minutes to reach school (i.e. late for school) on a randomly chosen day.[2]
2. In a month of \(30\) days, what is the expected number of days he will be late for school?[1]
3. Find the latest time he would have to reach the bus-stop outside his house so that the probability of him being late for school is less than \(5\)%.[4]
4. Find the probability that the average time taken to travel from the bus stop outside his house (including waiting for the bus) to school in \(30\) days is between \(15\) to \(20\) minutes.[2]