5 The duration, \(X\) minutes, of a timetabled 1-hour lesson may be assumed to be normally distributed with mean 54 and standard deviation 2. The duration, \(Y\) minutes, of a timetabled \(1 \frac { 1 } { 2 }\)-hour lesson may be assumed to be normally distributed with mean 83 and standard deviation 3. Assuming the durations of lessons to be independent, determine the probability that the total duration of a random sample of three 1 -hour lessons is less than the total duration of a random sample of two \(1 \frac { 1 } { 2 }\)-hour lessons.
(7 marks)