N1983 P2 Q11

Observations of large number of train journeys from \(A\) and \(B\) have shown that the mean departure time is \(1000\) hrs, and that the departure times are normally distributed with a standard deviation of \(3\) minutes. The scheduled time of arrival at \(B\) is \(1400\) hrs, but the time the journey takes is normally distributed with a mean of \(234\) minutes and a standard deviation of \(4\) minutes. Find

1. the distribution of arrival times,
2. the probability that the train will be late,
3. the probability that in \(10\) consecutive journeys the train will be late at least once.

Also find, to the nearest minute, the earliest time to which the scheduled arrival time should be changed so that the probability that train will be late is less than \(0.01\).

Modelling assumptions for this exercise: departure time and journey duration are independent, and the ten journeys are independent and follow the same stated model.