Decision trees and expected monetary valueEdexcel A-Level Further Maths: Subtopic test

1. 1
   A florist must decide whether to order a large or a small stock of roses for a weekend. The weather will be fine with probability 0.6 or poor with probability 0.4. A large stock gives a profit of £800 if the weather is fine and £200 if it is poor. A small stock gives a profit of £500 if fine and £350 if poor.

   (a)

   Find the expected monetary value (EMV) of ordering a large stock.

   [1 mark]

   • A£440
   • B£560
   • C£500
   • D£1000

   (b)

   To maximise EMV, which course of action should the florist choose?

   [1 mark]

   • ASmall stock, because its worst outcome is better
   • BLarge stock, because its best outcome is £800
   • CSmall stock, because its EMV is £440
   • DLarge stock, because its EMV of £560 is greater than £440

   (c)

   The probability of fine weather is $p$ instead of $0.6$. Find the value of $p$ for which the two stock sizes have the same EMV.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A company must choose between two projects. Project X gives a certain profit of £40 000. Project Y gives a profit of £100 000 with probability 0.5, and otherwise a loss of £20 000.

   (a)

   Find the EMV of Project Y.

   [1 mark]

   • A£60 000
   • B£80 000
   • C£40 000
   • D£50 000

   (b)

   Based on EMV alone, which statement is correct?

   [1 mark]

   • AThe company is indifferent between X and Y, as both have an EMV of £40 000
   • BThe company should choose X, as it has no risk
   • CThe company should choose Y, as it could make £100 000
   • DThe company should choose neither, as Y could make a loss

   (c)

   The directors assign a utility of 0 to a loss of £20 000, 0.8 to a profit of £40 000 and 1 to a profit of £100 000. Calculate the expected utility of each project and state which the company should choose.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A software firm is deciding whether to launch a new app. Without any research, the probability that the app succeeds is 0.5. A success gives a profit of £120 000 and a failure gives a loss of £50 000; not launching gives £0. Alternatively, the firm can first pay £10 000 for a market survey. The survey result is favourable with probability 0.6, and then the probability of success is 0.75; otherwise it is unfavourable, and then the probability of success is 0.125. The profits and losses above do not include the cost of the survey.

   (a)

   Find the EMV of launching the app after an unfavourable survey result, and state the decision the firm should then make.

   [3 marks]

   (b)

   Determine whether the firm should commission the survey. Show the value at each node you use.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A mining company must decide whether to drill at a site. Drilling costs £2 million. If it drills, there is a 0.3 probability of a major find with revenue £12 million, a 0.45 probability of a minor find with revenue £4 million, and a 0.25 probability of finding nothing (revenue £0). Alternatively, the company can sell its drilling rights for a certain £1.5 million.

   (a)

   (i) State the pay-off at each end node of the drilling branch.
   (ii) Find the EMV of drilling and the value at the decision node.
   (iii) Explain why the value at the decision node is the larger of two values, while the value at the chance node is a weighted average.

   [6 marks]

   (b)

   The company's directors are risk averse. They assign a utility of 0 to $-$£2 million, 0.6 to £1.5 million, 0.65 to £2 million and 1 to £10 million. Evaluate whether the company should drill, using expected utility, and compare your conclusion with the EMV decision.

   [6 marks]

   Total for question 4: 12 marks

End of questions