2 The operator of a section of motorway toll road records its weekly takings according to the types of vehicles using the motorway. For purposes of charging, there are three types of vehicle: cars, coaches, lorries. The weekly takings (in thousands of pounds) for each type are assumed to be Normally distributed. These distributions are independent of each other and are summarised in the table.
| Vehicle type | Mean | Standard deviation |
| Cars | 60.2 | 5.2 |
| Coaches | 33.9 | 6.3 |
| Lorries | 52.4 | 4.9 |
- Find the probability that the weekly takings for coaches are less than \(\pounds 40000\).
- Find the probability that the weekly takings for lorries exceed the weekly takings for cars.
- Find the probability that over a 4 -week period the total takings for cars exceed \(\pounds 225000\). What assumption must be made about the four weeks?
- Each week the operator allocates part of the takings for repairs. This is determined for each type of vehicle according to estimates of the long-term damage caused. It is calculated as follows: \(5 \%\) of takings for cars, \(10 \%\) for coaches and \(20 \%\) for lorries. Find the probability that in any given week the total amount allocated for repairs will exceed \(\pounds 20000\).