3 Turn over P72118A 3. The table below shows the transport options, usual travel times, possible delay times and corresponding probabilities of delay for a journey. All times are in minutes. Transport option Usual travel time Possible delay time Probability of delay Car 52 10 0.10 25 0.02 Train 45 15 0.05 25 0.03 Coach 55 5 0.05 15 0.01 (a) Draw a decision tree to model the transport options and the possible outcomes. (5) (b) State the minimum expected travel time and the corresponding transport option indicated by the decision tree. (2) (Total for Question 3 is 7 marks) 4 P72118A 4. C1 20 33 41 30 17 6 14 22 53 75 24 8 4 14 30 A B C D S T E F G C1 C2 C2 20 29 41 25 15 6 14 22 42 72 19 5 4 9 30 10 12 Figure 1 Figure 1 shows a capacitated, directed network of pipes. The uncircled number on each arc represents the capacity of the corresponding pipe. The numbers in circles represent an initial flow. (a) List the saturated arcs. (1) (b) State the value of the initial flow. (1) (c) Explain why arc FT cannot be full to capacity. (1) (d) State the capacity of cut C1 and the capacity of cut C2 (2) (e) By inspection find one flow‑augmenting route to increase the flow by three units. You must state your route. (1) (f) Prove that, once the flow‑augmenting route found in part (e) has been applied, the flow is maximal. (3) (Total for Question 4 is 9 marks)