A2 June 2024 Q5
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 £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 |
| Scheme | Marks | AO |
|---|---|---|
Note arcs must show delay time and probability | M1 A1 M1 A1 M1 A1 | 3.3 1.1b 3.4 1.1b 3.4 1.1b |
| (6) |
Notes
(a) Condone additional arcs seen with 0 probability (plane 3 hours, train 3 and 24 hours, coach 24 hours)
M1: tree diagram with at least eight end pay-offs, one decision node and three chance nodes (condone missing triangles from end pay offs and incorrect shapes for decision and chance nodes)
A1: correct structure of tree diagram with the non-zero probability 11 arcs labelled correctly (including probabilities)
M1: at least three end-pay offs consistent with their stated probabilities (must include ticket price, cost for travel time and cost for delay – may be implied by correct values); at least eight attempted
A1: all eleven end-pay offs correct including triangles and no incorrect extra (condone if not fully simplified) (values may be negative as costs)
M1: all three chance nodes attempted with their probabilities
A1: CAO for chance and decision nodes including double line through inferior options – must have the correct shapes
| Scheme | Marks | AO |
|---|---|---|
| Travel option is Train | dB1 | 2.2a |
| (1) |
Notes
dB1: deduction of correct travel option (dependent on all method marks earned in (a))
| Scheme | Marks | AO |
|---|---|---|
| Utility values are 7.2427…, 7.2820…, 7.3282… | M1 A1 | 1.1b 1.1b |
| Therefore, the travel option with the best expected utility is Plane | A1 | 2.2a |
| (3) | ||
| (10 marks) |
Notes
M1: At least one (of the three) utility values correct (FT their end pay offs)
A1: At least two correct
A1: Correct travel option (Plane) together with all three correct values (to at least 3 sf – rounded or truncated)