A2 October 2020 Q2

EdexcelCurrent spec7 marksDecision Analysis

2. Jenny can choose one of three options, A, B or C, when playing a game. The profit, in pounds, associated with each outcome and their corresponding probabilities are shown on the decision tree in Figure 1.

Decision tree: a decision node with branches A, B and C, each to a chance node. A: probability 0.6 to 350 and 0.4 to −140. B: probability 0.75 to 260 and 0.25 to −190. C: probability 0.8 to 220 and 0.2 to −230 — Figure 1

(a) Calculate the optimal EMV to determine Jenny’s best course of action. You must make your working clear. (3)

For a profit of £\(x\), Jenny’s utility is given by \(1 - \mathrm{e}^{-\frac{x}{400}}\)

(b) Using expected utility as the criterion for the best course of action, determine what Jenny should do now to maximise her profit. You must make your working clear. (4)

Mark scheme (a)

Scheme	Marks	AO
EMV for A is \(0.6(350) + 0.4(-140) = 154\) EMV for B is \(0.75(260) + 0.25(-190) = 147.5\) EMV for C is \(0.8(220) + 0.2(-230) = 130\)	M1 A1	3.4 1.1b
The optimal EMV is £154, which makes option A the best choice using the EMV criterion	A1	2.2a
	(3)

Notes

M1: Correct method for calculation EMV for either A, B or C

A1: Correct values of EMV for A, B and C

A1: Correct deduction of optimal EMV (dependent on all three correct EMVs)

Mark scheme (b)

Scheme	Marks	AO
\(u(350) = 0.5831379803\ldots,\ u(-140) = -0.4190675486\ldots\) \(u(260) = 0.4779542232\ldots,\ u(-190) = -0.6080141975\ldots\) \(u(220) = 0.4230501896\ldots,\ u(-230) = -0.7771305269\ldots\)	M1	3.4
Calculate all three expected utilities: A is \(0.6(0.583\ldots) + 0.4(-0.419\ldots) = 0.1822557\ldots\) B is \(0.75(0.477\ldots) + 0.25(-0.608\ldots) = 0.2064621\ldots\) C is \(0.8(0.423\ldots) + 0.2(-0.777\ldots) = 0.1830140\ldots\)	DM1 A1	1.1b 1.1b
The optimal expected utility is 0.206 utils, which makes option B the best choice using expected utility as the criterion	A1	2.2a
	(4)
	(7 marks)

Notes

M1: Uses the correct utility function to replace each pay-off with the corresponding utility

DM1: Calculate all three expected utilities using correct probability values from (a)

A1: At least 2 expected utilities correct (correct to at least 2 decimal places)

A1: Correct deduction of optimal expected utility