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Strategic Cost Management · Decision Making using Probability

Decision Trees: Decision Nodes, Chance Nodes and Rollback Method

Updated 11 October 2026 · Fact-checked

A decision tree is a diagram of sequential choices and uncertain outcomes. You draw decision nodes (squares) and chance nodes (circles), then roll back from right to left. At each chance node take the expected value; at each decision node pick the best option. Deduct costs on the way.

Understand Decision Trees

A decision tree maps a problem in which you must choose now, see what happens, and then choose again. It is used when outcomes are uncertain and each choice affects what you can do later. Examples are launching a product, building a plant of one size or another, or testing before full production.

There are two kinds of node. A decision node (drawn as a square) is a point where you choose. Each branch from it is an option. A chance node (drawn as a circle) is a point where nature decides. Each branch from it is an outcome with a probability, and the probabilities on branches leaving one chance node must add up to 1.

You solve a tree by rolling back. Start at the far right, where the final payoffs are. Move left. At a chance node, replace the node with its expected monetary value (EMV), which is the sum of probability × payoff over its branches. At a decision node, compare the values of the options (after deducting the cost of each option) and keep the best. Cross out the rejected branches. Continue until you reach the first decision.

The method works because later decisions are made after the uncertainty is resolved. So you must value the later decision first, and then ask what the earlier choice is worth. Solving left to right gives wrong answers.

The result is a strategy, not only a number. It tells you the best first choice and what to do after each outcome. EMV is a long-run average. It is appropriate when the decision is repeated or the stakes are small relative to the firm. It does not capture attitude to risk.

Key rules to remember

EMV at a chance node
EMV = Σ (probability × payoff of each branch)
Probabilities on branches from one chance node must total 1.
Value at a decision node (profit)
Value = highest of (EMV of option − cost of option)
For a cost-minimising problem, choose the lowest expected cost instead.
Net value of a strategy
Net EMV = EMV of outcomes − cost incurred to reach that point
Deduct costs along the path once, at the node where they are incurred.
Rollback rule
Work right to left: chance node → expected value; decision node → best option
Cross out rejected branches so the chosen path is clear.

How to solve Decision Trees questions

Use this order for any decision tree question. It keeps the working clean and shows the examiner each step.

  1. 1Read the problem and list the decisions, the uncertain events, their probabilities and all costs and payoffs.
  2. 2Draw the tree from left to right in time order. Use a square for each decision and a circle for each chance event. Write the probability on each chance branch and the cost on each option branch.
  3. 3Put the final payoff at the end of every path. Check that payoffs are on the same basis, for example total profit before deducting the option cost.
  4. 4Check that probabilities at each chance node add up to 1.
  5. 5Roll back from the right. Compute the EMV at each chance node and write it above the node.
  6. 6At each decision node, subtract the option cost from the EMV of each branch, compare, and cross out the inferior branch.
  7. 7Continue left to the first decision and state the recommended strategy in words, with the expected value.
  8. 8If asked, comment on risk, the range of outcomes or the value of information.

Quickest way: Right-to-left shortcut with net values

When to use it: Use it when the tree is large and time is short, and when costs are given on option branches.

  1. Sketch the tree roughly. Mark only squares, circles, probabilities and payoffs.
  2. Start with the last chance nodes and write EMV beside each.
  3. At the next decision node, write the net value for each option as EMV minus cost, and tick the larger.
  4. Carry only the ticked value to the left. Do not recompute rejected branches.
  5. Write the strategy in one sentence: first choice, then the action after each outcome.

Common mistakes in Decision Trees

  • Solving from left to right.

    The tree is read like a story, so students calculate in time order.

    Fix: Always start at the rightmost payoffs and move left. Later choices must be valued first.

  • Forgetting to deduct the cost of an option.

    The cost sits on the branch, and students compare only the EMVs.

    Fix: At each decision node compute EMV minus cost for every option before comparing.

  • Deducting the same cost twice.

    The cost is subtracted at the chance node and again at the decision node.

    Fix: Deduct each cost once, at the decision node where the option is chosen.

  • Probabilities that do not add up to 1 at a node.

    Students copy a probability from another node or miss the complement.

    Fix: Add the probabilities at every chance node before computing EMV.

  • Giving only a number and no strategy.

    Students stop once the first node is valued.

    Fix: State the first decision and what to do after each outcome, then give the expected value.

  • Treating rejected branches as part of the answer.

    The values are not crossed out and get mixed into later nodes.

    Fix: Strike through inferior branches and carry forward only the best value.

Worked examples

Example 1

Sundaram Foods Ltd must choose between launching a new snack nationally or doing nothing (payoff ₹0). National launch costs ₹40,00,000. Success (probability 0.6) gives a profit of ₹1,00,00,000 before launch cost; failure (0.4) gives ₹20,00,000 before launch cost. Which option should it choose on EMV?

Show the solution
  1. Decision: launch or do nothing.
  2. Launch, chance node: EMV of inflows = 0.6 × 1,00,00,000 + 0.4 × 20,00,000 = 60,00,000 + 8,00,000 = ₹68,00,000.
  3. Deduct launch cost: 68,00,000 − 40,00,000 = ₹28,00,000.
  4. Do nothing = ₹0.
  5. Compare: 28,00,000 > 0.

Answer: Launch nationally. The expected net value is ₹28,00,000.

Example 2

Kaveri Engineering can first spend ₹5,00,000 on a pilot run. The pilot is favourable with probability 0.7. After a favourable pilot, full production succeeds with probability 0.8; after an unfavourable pilot, it succeeds with probability 0.1. Full production costs ₹45,00,000 and gives revenue of ₹80,00,000 on success or ₹40,00,000 on failure. The firm can also stop (payoff ₹0). Without the pilot, the firm can go straight to full production on the same terms. Its overall success probability is then 0.7 × 0.8 + 0.3 × 0.1 = 0.59. Which strategy has the higher expected value?

Show the solution
  1. Pilot, favourable, full production: EMV of revenue = 0.8 × 80,00,000 + 0.2 × 40,00,000 = 64,00,000 + 8,00,000 = ₹72,00,000.
  2. Net of production cost = 72,00,000 − 45,00,000 = ₹27,00,000. Stop = 0. So produce; value = ₹27,00,000.
  3. Pilot, unfavourable, full production: EMV of revenue = 0.1 × 80,00,000 + 0.9 × 40,00,000 = 8,00,000 + 36,00,000 = ₹44,00,000. Net = 44,00,000 − 45,00,000 = −₹1,00,000. This is below the stop payoff of 0, so stop; value = ₹0.
  4. Chance node after pilot: 0.7 × 27,00,000 + 0.3 × 0 = ₹18,90,000.
  5. Net of pilot cost: 18,90,000 − 5,00,000 = ₹13,90,000.
  6. No pilot, produce: EMV of revenue = 0.59 × 80,00,000 + 0.41 × 40,00,000 = 47,20,000 + 16,40,000 = ₹63,60,000. Net = 63,60,000 − 45,00,000 = ₹18,60,000. Stop = 0, so produce.
  7. Compare: 18,60,000 vs 13,90,000.

Answer: Go straight to full production without the pilot. Its expected value of ₹18,60,000 exceeds ₹13,90,000 for the pilot strategy. The pilot is not worth its ₹5,00,000 cost, because it can add at most ₹30,000 of value (it only helps by avoiding a ₹1,00,000 expected loss in the unfavourable case, which has probability 0.3).

Exam tips

  • Draw the tree neatly even in a numerical question. Marks are often given for the correct structure and for each node value.
  • Label every node value and cross out rejected branches. This helps the examiner award method marks even if you make an arithmetic slip.
  • Always finish with a recommendation in words, including the first decision and the follow-up actions.
  • In MCQs, check whether costs are already netted in the payoffs before deducting them again.
  • If the question asks about perfect information or risk, link your tree result to those ideas briefly after the calculation.

Practice questions from Decision Making using Probability

Decision Trees in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Decision Trees: frequently asked questions

What is the difference between a decision node and a chance node?

A decision node is a point where you choose among options, drawn as a square. A chance node is a point where an uncertain event occurs, drawn as a circle, with probabilities on its branches. You pick the best branch at a decision node and take the expected value at a chance node.

Why do we roll back from right to left?

Later decisions depend on what happens earlier, and their values must be known before you can judge the earlier choice. Starting at the payoffs and moving left values each later decision first. This gives the best overall strategy.

How do I treat costs in a decision tree?

Put each cost on the option branch where it is incurred. At the decision node, subtract it from the EMV of that option before comparing. Deduct each cost only once.

Is the highest EMV always the right choice?

It is the best choice on an expected-value basis. EMV is a long-run average, so it ignores how risky the spread of outcomes is. Where one bad outcome could be very costly, a comment on risk is also expected.