Further Maths Decision analysis Practice Questions

Free Further Maths Decision analysis practice questions with full step-by-step worked solutions. Covers decision-analysis, decision-tree, expected-monetary-value, chance-node. Practise exam-style problems and check your method.

decision-analysisdecision-treeexpected-monetary-valuechance-nodefold-backaveraging-out
Further Maths70 questionsStep-by-step solutions
Question 1
2 markseasy
A company faces a decision under uncertainty, modelled by the decision tree set out here. The tree is set out in words as follows. The root is the decision node D1D1. At the decision node D1D1 the options are: choose Harbour\text{Harbour} to reach C1C1; choose Summit\text{Summit} to reach C2C2. At the chance node C1C1 the outcomes are: with probability 23\frac{2}{3} reach −29-29; with probability 13\frac{1}{3} reach 5050. At the chance node C2C2 the outcomes are: with probability 25\frac{2}{5} reach 3333; with probability 35\frac{3}{5} reach 213213. At a chance node the expected monetary value is the sum, over the branches, of probability multiplied by the value reached along that branch; at a decision node the value is the largest option value, where an option's value is the value it leads to minus any cost of taking it. The tree is folded back from the leaves to the root. Find the expected monetary value at the chance node C1C1.
Show worked solution

Worked solution

  1. Set out the structure of the decision tree

    root D1, 1 decision node(s), 2 chance node(s)\text{root }D1,\ 1\text{ decision node(s)},\ 2\text{ chance node(s)}

    Square decision nodes are folded back by choosing the best option; circular chance nodes are averaged out; work from the leaves to the root.

  2. Average out at the chance node C1C1

    EMV(C1)=23×(−29)+13×50=−83\text{EMV}(C1)=\frac{2}{3}\times (-29) + \frac{1}{3}\times 50=-\frac{8}{3}

    The EMV at C1C1 is the sum of probability times the value reached along each branch.

  3. State the expected monetary value at C1C1

    EMV(C1)=−83\text{EMV}(C1)=-\frac{8}{3}

    This is the required expected monetary value at the chance node.

Answer
−83-\frac{8}{3}
Question 2
2 markseasy
A firm must select a single course of action in the face of an uncertain outcome. A choice must be made between the actions Beacon\text{Beacon}, Lumen\text{Lumen}. Exactly one of the states S1S1, S2S2 will occur, with probabilities P(S1)=35P(S1)=\frac{3}{5}, P(S2)=25P(S2)=\frac{2}{5}. The payoffs are as follows: action Beacon\text{Beacon} yields 188188, 11 under S1S1, S2S2 respectively; action Lumen\text{Lumen} yields −31-31, 182182 under S1S1, S2S2 respectively. Find the greatest expected monetary value obtainable without further information.
Show worked solution

Worked solution

  1. Set out the actions, states and probabilities

    actions=2, states=2\text{actions}=2,\ \text{states}=2

    Each action is evaluated against every state of nature, weighted by the probability of that state.

  2. Find the expected monetary value of action Beacon\text{Beacon}

    EMV(Beacon)=35×188+25×1=5665\text{EMV}(\text{Beacon})=\frac{3}{5}\times 188 + \frac{2}{5}\times 1=\frac{566}{5}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  3. Find the expected monetary value of action Lumen\text{Lumen}

    EMV(Lumen)=35×(−31)+25×182=2715\text{EMV}(\text{Lumen})=\frac{3}{5}\times (-31) + \frac{2}{5}\times 182=\frac{271}{5}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  4. State the greatest expected monetary value

    max⁡aEMV=5665 (action Beacon)\max_a\text{EMV}=\frac{566}{5}\ (\text{action }\text{Beacon})

    Without further information the decision-maker takes the action of greatest EMV.

Answer
5665\frac{566}{5}
Question 3
4 marksintermediate
A company faces a decision under uncertainty, modelled by the decision tree set out here. The tree is set out in words as follows. The root is the decision node D1D1. At the decision node D1D1 the options are: choose Ridge\text{Ridge} to reach C1C1; choose Meadow\text{Meadow} to reach C2C2; choose Radial\text{Radial} to reach C3C3; choose Dynamo\text{Dynamo} to reach C4C4; choose Alpha\text{Alpha} to reach C5C5. At the chance node C1C1 the outcomes are: with probability 712\frac{7}{12} reach 8989; with probability 512\frac{5}{12} reach 2727. At the chance node C2C2 the outcomes are: with probability 710\frac{7}{10} reach 8484; with probability 310\frac{3}{10} reach 2626. At the chance node C3C3 the outcomes are: with probability 15\frac{1}{5} reach 103103; with probability 45\frac{4}{5} reach 209209. At the chance node C4C4 the outcomes are: with probability 12\frac{1}{2} reach 139139; with probability 12\frac{1}{2} reach 5555. At the chance node C5C5 the outcomes are: with probability 712\frac{7}{12} reach −35-35; with probability 512\frac{5}{12} reach 220220. The decision-maker assigns the following utilities to the possible payoffs: U(−35)=0U(-35)=0, U(26)=10U(26)=10, U(27)=19U(27)=19, U(55)=27U(55)=27, U(84)=34U(84)=34, U(89)=40U(89)=40, U(103)=45U(103)=45, U(139)=49U(139)=49, U(209)=52U(209)=52, U(220)=54U(220)=54. The decision-maker judges options not by money but by utility. The expected utility of an option is the sum, over the branches, of probability multiplied by the utility of the payoff reached. Using the utilities given, which option gives the greatest expected utility?
Show worked solution

Worked solution

  1. Set out the structure of the decision tree

    root D1, 1 decision node(s), 5 chance node(s)\text{root }D1,\ 1\text{ decision node(s)},\ 5\text{ chance node(s)}

    Square decision nodes are folded back by choosing the best option; circular chance nodes are averaged out; work from the leaves to the root.

  2. Take the expected utility at the chance node C1C1

    EU(C1)=712×40+512×19=1254\text{EU}(C1)=\frac{7}{12}\times 40 + \frac{5}{12}\times 19=\frac{125}{4}

    The expected utility at C1C1 is the sum of probability times the utility of the payoff on each branch.

  3. Take the expected utility at the chance node C2C2

    EU(C2)=710×34+310×10=1345\text{EU}(C2)=\frac{7}{10}\times 34 + \frac{3}{10}\times 10=\frac{134}{5}

    The expected utility at C2C2 is the sum of probability times the utility of the payoff on each branch.

  4. Take the expected utility at the chance node C3C3

    EU(C3)=15×45+45×52=2535\text{EU}(C3)=\frac{1}{5}\times 45 + \frac{4}{5}\times 52=\frac{253}{5}

    The expected utility at C3C3 is the sum of probability times the utility of the payoff on each branch.

  5. Take the expected utility at the chance node C4C4

    EU(C4)=12×49+12×27=38\text{EU}(C4)=\frac{1}{2}\times 49 + \frac{1}{2}\times 27=38

    The expected utility at C4C4 is the sum of probability times the utility of the payoff on each branch.

  6. Take the expected utility at the chance node C5C5

    EU(C5)=712×0+512×54=452\text{EU}(C5)=\frac{7}{12}\times 0 + \frac{5}{12}\times 54=\frac{45}{2}

    The expected utility at C5C5 is the sum of probability times the utility of the payoff on each branch.

  7. Choose the option of greatest expected utility

    recommend Radial (EU =2535)\text{recommend }\text{Radial}\ (\text{EU }=\frac{253}{5})

    A risk-sensitive decision-maker selects the option with the largest expected utility.

Answer
Radial\text{Radial}
Question 4
6 markshard
A manager must choose one action now, before learning which state of nature will occur. A choice must be made between the actions Summit\text{Summit}, Dynamo\text{Dynamo}, Vantage\text{Vantage}, Jasper\text{Jasper}, Marble\text{Marble}. Exactly one of the states S1S1, S2S2, S3S3 will occur, with probabilities P(S1)=13P(S1)=\frac{1}{3}, P(S2)=512P(S2)=\frac{5}{12}, P(S3)=14P(S3)=\frac{1}{4}. The payoffs are as follows: action Summit\text{Summit} yields 2121, −3-3, −26-26 under S1S1, S2S2, S3S3 respectively; action Dynamo\text{Dynamo} yields 185185, 2424, −19-19 under S1S1, S2S2, S3S3 respectively; action Vantage\text{Vantage} yields −6-6, 130130, 4747 under S1S1, S2S2, S3S3 respectively; action Jasper\text{Jasper} yields 1313, 118118, 4848 under S1S1, S2S2, S3S3 respectively; action Marble\text{Marble} yields −12-12, 210210, 149149 under S1S1, S2S2, S3S3 respectively. Which action has the greatest expected monetary value?
Show worked solution

Worked solution

  1. Set out the actions, states and probabilities

    actions=5, states=3\text{actions}=5,\ \text{states}=3

    Each action is evaluated against every state of nature, weighted by the probability of that state.

  2. Find the expected monetary value of action Summit\text{Summit}

    EMV(Summit)=13×21+512×(−3)+14×(−26)=−34\text{EMV}(\text{Summit})=\frac{1}{3}\times 21 + \frac{5}{12}\times (-3) + \frac{1}{4}\times (-26)=-\frac{3}{4}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  3. Find the expected monetary value of action Dynamo\text{Dynamo}

    EMV(Dynamo)=13×185+512×24+14×(−19)=80312\text{EMV}(\text{Dynamo})=\frac{1}{3}\times 185 + \frac{5}{12}\times 24 + \frac{1}{4}\times (-19)=\frac{803}{12}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  4. Find the expected monetary value of action Vantage\text{Vantage}

    EMV(Vantage)=13×(−6)+512×130+14×47=76712\text{EMV}(\text{Vantage})=\frac{1}{3}\times (-6) + \frac{5}{12}\times 130 + \frac{1}{4}\times 47=\frac{767}{12}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  5. Find the expected monetary value of action Jasper\text{Jasper}

    EMV(Jasper)=13×13+512×118+14×48=1312\text{EMV}(\text{Jasper})=\frac{1}{3}\times 13 + \frac{5}{12}\times 118 + \frac{1}{4}\times 48=\frac{131}{2}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  6. Find the expected monetary value of action Marble\text{Marble}

    EMV(Marble)=13×(−12)+512×210+14×149=4834\text{EMV}(\text{Marble})=\frac{1}{3}\times (-12) + \frac{5}{12}\times 210 + \frac{1}{4}\times 149=\frac{483}{4}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  7. Check the state probabilities total 11

    13+512+14=1\frac{1}{3} + \frac{5}{12} + \frac{1}{4}=1

    Exactly one state occurs, so the probabilities must add up to 11.

  8. List the expected monetary values of the actions

    EMV(Summit)=−34, EMV(Dynamo)=80312, EMV(Vantage)=76712, EMV(Jasper)=1312, EMV(Marble)=4834\text{EMV}(\text{Summit})=-\frac{3}{4},\ \text{EMV}(\text{Dynamo})=\frac{803}{12},\ \text{EMV}(\text{Vantage})=\frac{767}{12},\ \text{EMV}(\text{Jasper})=\frac{131}{2},\ \text{EMV}(\text{Marble})=\frac{483}{4}

    Each action has one EMV; the decision-maker without information takes the largest.

  9. Identify the best action without further information

    best=Marble (EMV =4834)\text{best}=\text{Marble}\ (\text{EMV }=\frac{483}{4})

    This is the action a decision-maker would take knowing only the probabilities.

  10. Select the action of greatest expected monetary value

    recommend Marble (EMV =4834)\text{recommend }\text{Marble}\ (\text{EMV }=\frac{483}{4})

    The action with the largest EMV is the recommended choice.

Answer
Marble\text{Marble}
Question 5
9 markschallenging
A decision-maker must commit to exactly one action while the future state of the market is uncertain. A choice must be made between the actions Forge\text{Forge}, Marble\text{Marble}, Ingot\text{Ingot}, Willow\text{Willow}, Sable\text{Sable}. Exactly one of the states S1S1, S2S2, S3S3, S4S4 will occur, with probabilities P(S1)=110P(S1)=\frac{1}{10}, P(S2)=35P(S2)=\frac{3}{5}, P(S3)=15P(S3)=\frac{1}{5}, P(S4)=110P(S4)=\frac{1}{10}. The payoffs are as follows: action Forge\text{Forge} yields 2626, 100100, 185185, 99 under S1S1, S2S2, S3S3, S4S4 respectively; action Marble\text{Marble} yields 1313, −13-13, 219219, 131131 under S1S1, S2S2, S3S3, S4S4 respectively; action Ingot\text{Ingot} yields 157157, 132132, 144144, 139139 under S1S1, S2S2, S3S3, S4S4 respectively; action Willow\text{Willow} yields 8181, 186186, 136136, 180180 under S1S1, S2S2, S3S3, S4S4 respectively; action Sable\text{Sable} yields 163163, 126126, 3232, 2020 under S1S1, S2S2, S3S3, S4S4 respectively. Which action has the greatest expected monetary value?
Show worked solution

Worked solution

  1. Set out the actions, states and probabilities

    actions=5, states=4\text{actions}=5,\ \text{states}=4

    Each action is evaluated against every state of nature, weighted by the probability of that state.

  2. Find the expected monetary value of action Forge\text{Forge}

    EMV(Forge)=110×26+35×100+15×185+110×9=2012\text{EMV}(\text{Forge})=\frac{1}{10}\times 26 + \frac{3}{5}\times 100 + \frac{1}{5}\times 185 + \frac{1}{10}\times 9=\frac{201}{2}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  3. Find the expected monetary value of action Marble\text{Marble}

    EMV(Marble)=110×13+35×(−13)+15×219+110×131=2525\text{EMV}(\text{Marble})=\frac{1}{10}\times 13 + \frac{3}{5}\times (-13) + \frac{1}{5}\times 219 + \frac{1}{10}\times 131=\frac{252}{5}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  4. Find the expected monetary value of action Ingot\text{Ingot}

    EMV(Ingot)=110×157+35×132+15×144+110×139=6885\text{EMV}(\text{Ingot})=\frac{1}{10}\times 157 + \frac{3}{5}\times 132 + \frac{1}{5}\times 144 + \frac{1}{10}\times 139=\frac{688}{5}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  5. Find the expected monetary value of action Willow\text{Willow}

    EMV(Willow)=110×81+35×186+15×136+110×180=164910\text{EMV}(\text{Willow})=\frac{1}{10}\times 81 + \frac{3}{5}\times 186 + \frac{1}{5}\times 136 + \frac{1}{10}\times 180=\frac{1649}{10}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  6. Find the expected monetary value of action Sable\text{Sable}

    EMV(Sable)=110×163+35×126+15×32+110×20=100310\text{EMV}(\text{Sable})=\frac{1}{10}\times 163 + \frac{3}{5}\times 126 + \frac{1}{5}\times 32 + \frac{1}{10}\times 20=\frac{1003}{10}

    The EMV of an action is the probability-weighted sum of its payoffs across the states.

  7. Check the state probabilities total 11

    110+35+15+110=1\frac{1}{10} + \frac{3}{5} + \frac{1}{5} + \frac{1}{10}=1

    Exactly one state occurs, so the probabilities must add up to 11.

  8. List the expected monetary values of the actions

    EMV(Forge)=2012, EMV(Marble)=2525, EMV(Ingot)=6885, EMV(Willow)=164910, EMV(Sable)=100310\text{EMV}(\text{Forge})=\frac{201}{2},\ \text{EMV}(\text{Marble})=\frac{252}{5},\ \text{EMV}(\text{Ingot})=\frac{688}{5},\ \text{EMV}(\text{Willow})=\frac{1649}{10},\ \text{EMV}(\text{Sable})=\frac{1003}{10}

    Each action has one EMV; the decision-maker without information takes the largest.

  9. Identify the best action without further information

    best=Willow (EMV =164910)\text{best}=\text{Willow}\ (\text{EMV }=\frac{1649}{10})

    This is the action a decision-maker would take knowing only the probabilities.

  10. Recall the meaning of EVPI

    EVPI=EVwPI−best EMV\text{EVPI}=\text{EVwPI}-\text{best EMV}

    The value of perfect information is what a perfect forecast would add over the best decision made without it.

  11. Note that EVPI cannot be negative

    EVwPI ≥ best EMV\text{EVwPI}\ \ge\ \text{best EMV}

    Perfect information can never make the decision-maker worse off, so the EVPI is at least zero.

  12. State how many states of nature there are

    states=4\text{states}=4

    Perfect information would reveal which of these states is going to occur.

  13. Find the best action if state S1S1 were known to occur

    S1: Sable (163)S1:\ \text{Sable}\ (163)

    Given state S1S1, the decision-maker would take the action with the largest payoff in that column.

  14. Find the best action if state S2S2 were known to occur

    S2: Willow (186)S2:\ \text{Willow}\ (186)

    Given state S2S2, the decision-maker would take the action with the largest payoff in that column.

  15. Select the action of greatest expected monetary value

    recommend Willow (EMV =164910)\text{recommend }\text{Willow}\ (\text{EMV }=\frac{1649}{10})

    The action with the largest EMV is the recommended choice.

Answer
Willow\text{Willow}

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