Decision analysis Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Decision analysis questions. See exactly how to solve problems on decision-analysis, decision-tree, expected-monetary-value, chance-node.

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.

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
An investor must decide between several courses of action, 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 Talon\text{Talon} to reach C1C1; choose Wander\text{Wander} to reach C2C2. At the chance node C1C1 the outcomes are: with probability 15\frac{1}{5} reach 116116; with probability 45\frac{4}{5} reach 1919. At the chance node C2C2 the outcomes are: with probability 710\frac{7}{10} reach −7-7; with probability 310\frac{3}{10} reach −32-32. 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.

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)=15×116+45×19=1925\text{EMV}(C1)=\frac{1}{5}\times 116 + \frac{4}{5}\times 19=\frac{192}{5}

    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)=1925\text{EMV}(C1)=\frac{192}{5}

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

Answer
1925\frac{192}{5}
Question 3
2 markseasy
A project team is choosing a plan 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 Quantum\text{Quantum} to reach C1C1; choose Ridge\text{Ridge} to reach C2C2. At the chance node C1C1 the outcomes are: with probability 25\frac{2}{5} reach 9797; with probability 35\frac{3}{5} reach 105105. At the chance node C2C2 the outcomes are: with probability 58\frac{5}{8} reach 187187; with probability 38\frac{3}{8} reach −28-28. 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.

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)=25×97+35×105=5095\text{EMV}(C1)=\frac{2}{5}\times 97 + \frac{3}{5}\times 105=\frac{509}{5}

    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)=5095\text{EMV}(C1)=\frac{509}{5}

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

Answer
5095\frac{509}{5}
Question 4
2 markseasy
A manufacturer must commit to one option now, 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 Pinnacle\text{Pinnacle} to reach C1C1; choose Alpha\text{Alpha} to reach C2C2. At the chance node C1C1 the outcomes are: with probability 12\frac{1}{2} reach 122122; with probability 12\frac{1}{2} reach 7171. At the chance node C2C2 the outcomes are: with probability 12\frac{1}{2} reach 203203; with probability 12\frac{1}{2} reach −11-11. 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.

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)=12×122+12×71=1932\text{EMV}(C1)=\frac{1}{2}\times 122 + \frac{1}{2}\times 71=\frac{193}{2}

    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)=1932\text{EMV}(C1)=\frac{193}{2}

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

Answer
1932\frac{193}{2}
Question 5
2 markseasy
A trader is weighing up several strategies, 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 Umber\text{Umber} to reach C1C1; choose Ingot\text{Ingot} to reach C2C2. At the chance node C1C1 the outcomes are: with probability 45\frac{4}{5} reach 108108; with probability 15\frac{1}{5} reach 187187. At the chance node C2C2 the outcomes are: with probability 310\frac{3}{10} reach 88; with probability 710\frac{7}{10} reach 3939. 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.

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)=45×108+15×187=6195\text{EMV}(C1)=\frac{4}{5}\times 108 + \frac{1}{5}\times 187=\frac{619}{5}

    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)=6195\text{EMV}(C1)=\frac{619}{5}

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

Answer
6195\frac{619}{5}

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