5. Sebastien needs to make a journey. He can choose between travelling by plane, by train or by coach.
Sebastien knows the exact costs of all three travel options, but he also wants to account for his travel time, including any possible delays.
The cost of Sebastien's time is \(\pounds 50\) per hour.
The table below shows the costs, the journey times (without delays), and the corresponding probabilities of delays, for each travel option.
| Cost of travel option | Journey time (in hours) without delays | Probability of a 1-hour delay | Probability of a 2-hour delay | Probability of a 3-hour delay | Probability of a 24-hour delay |
| Plane | £200 | 3 | 0.09 | 0.05 | 0 | 0.03 |
| Train | £130 | 5 | 0.07 | 0.03 | 0 | 0 |
| Coach | £70 | 6 | 0.15 | 0.1 | 0.05 | 0 |
- By drawing a decision tree, evaluate the EMV of the total cost of Sebastien's journey for each node of your tree.
- Hence state the travel option that minimises the EMV of the total cost of Sebastien's journey.
- A cube root utility function is applied to the total costs of each option. Determine the travel option with the best expected utility and state the corresponding value.