2024 DHS P2 Q9

Mr Hsu takes the train to office for work every weekday and is supposed to arrive at office by \(7.30\) a.m. On average, he reaches the train station at \(6.00\) a.m. His arrival time at the train station is normally distributed with a standard deviation of \(10\) minutes. Every morning, there are only two trains, each departing \(5\) minutes apart, with the first train departing at \(6:10\) a.m. sharp. The time taken for the train journey follows a normal distribution with mean \(60\) minutes and standard deviation \(4\) minutes.

After alighting from the train, the time taken to walk from the train station to his office follows a normal distribution with mean \(10\) minutes and standard deviation \(3\) minutes. Mr Hsu will be late for work if he misses the second train or arrives at office after \(7.30\) a.m. Assume that all travelling and waiting times are independent of each other.

1. On a randomly chosen day, given that Mr Hsu takes the first train, find the probability that he is late for work.[2]
2. On a randomly chosen day, show that the probability that Mr Hsu is late for work is \(0.101\).[3]
3. On a randomly chosen day, Mr Hsu has a briefing to attend before work. Given that he takes the first train, find the earliest starting time of the briefing (correct to the nearest minutes) for which the probability that he is late for the briefing is not more than \(0.1\).[2]

According to Mr Hsu's office policy, employees who are late for work will face a pay deduction of \(D\%\), where \(D\) is calculated as \(5\) times the number of days the employee is late in a month. Given that there are \(20\) workdays in a month,

4. find the probability that Mr Hsu receives between \(60\%\) and \(80\%\) inclusive, of his salary in a randomly chosen month.[3]