2024 DHS H1 Q11

IB AA SL: core and optional content

Core: the final expected-day count using the probability supplied in part(b). Optional: deriving lateness probabilities and the briefing time using the sum of journey and walking times.

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, find the probability that Mr Hsu takes the first train and 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 scheduled on a whole minute for which the probability that he is late for the briefing is not more than \(0.1\).[2]
4. In a randomly chosen year of \(260\) work days, find the expected number of days that Mr Hsu arrive late for work.[2]