5 Roger is an active retired lecturer. Each day after breakfast, he decides whether the weather for that day is going to be fine ( \(F\) ), dull ( \(D\) ) or wet ( \(W\) ). He then decides on only one of four activities for the day: cycling ( \(C\) ), gardening ( \(G\) ), shopping ( \(S\) ) or relaxing \(( R )\). His decisions from day to day may be assumed to be independent.
The table shows Roger's probabilities for each combination of weather and activity.
| \multirow{2}{*}{} | Weather |
| | Fine ( \(F\) ) | Dull ( \(D\) ) | Wet ( \(\boldsymbol { W }\) ) |
| \multirow{4}{*}{Activity} | Cycling ( \(\boldsymbol { C }\) ) | 0.30 | 0.10 | 0 |
| Gardening ( \(\boldsymbol { G }\) ) | 0.25 | 0.05 | 0 |
| Shopping ( \(\boldsymbol { S }\) ) | 0 | 0.10 | 0.05 |
| Relaxing ( \(\boldsymbol { R }\) ) | 0 | 0.05 | 0.10 |
- Find the probability that, on a particular day, Roger decided:
- that it was going to be fine and that he would go cycling;
- on either gardening or shopping;
- to go cycling, given that he had decided that it was going to be fine;
- not to relax, given that he had decided that it was going to be dull;
- that it was going to be fine, given that he did not go cycling.
- Calculate the probability that, on a particular Saturday and Sunday, Roger decided that it was going to be fine and decided on the same activity for both days.