## **Learning Outcomes**

After reading this article, you should be able to apply probability and decision tree techniques to business decisions involving uncertainty. You will understand sequential (multi-stage) decisions, how to assign and use conditional probabilities, and how to perform rollback analysis to identify optimal options. By achieving proficiency in these methods, you will be able to advise management effectively in scenarios requiring expected value or risk-based approaches.

## **ACCA Advanced Performance Management (APM) Syllabus**

For ACCA Advanced Performance Management (APM), you are required to understand how probability and decision trees support better decision-making under uncertainty. In particular, you must cover:

- The principles of probability in performance management and scenario planning
- The construction and interpretation of decision trees, including payoff calculations
- The evaluation of sequential and multi-stage decisions using decision trees
- The application of rollback analysis to determine optimal decisions
- Awareness of risk attitudes and the implications for decision-making

## **Test Your Knowledge**

_Attempt these questions before reading this article. If you find some difficult or cannot remember the answers, remember to look more closely at that area during your revision._

1. What does "rollback analysis" refer to when using a decision tree?
2. In a sequential decision, how do you account for the effect of one decision on the probabilities of subsequent events?
3. True or false? The sum of probabilities on any set of mutually exclusive outcomes must be less than 1.
4. If a project has a 60% chance of success (payoff \$100,000) and a 40% chance of failure (payoff \$0), what is the expected value?
5. Why is it important to consider conditional probabilities in sequential decisions?

## **Introduction**

Probability and decision trees are essential analytical tools used in performance management when choices must be made under uncertainty. These methods enable you to structure complex, multi-stage decisions by mapping out each possible course of action and the likelihood and outcome of each scenario. Sequential decisions—where one choice influences the next set of available options—require special care, as the probability of each event may be affected by earlier decisions and outcomes.

Decision trees allow you to visually lay out all alternative actions, chance events, and payoffs, so you can systematically identify the sequence that maximises expected value or best meets other strategic objectives. The "rollback" technique then enables you to analyse the decision tree from right to left, ensuring at each step that you choose the action yielding the best outcome in light of future uncertainties.

> **Key Term: probability**
> The likelihood or chance of an event occurring, expressed as a value between 0 (impossible) and 1 (certain).
>
> **Key Term: decision tree**
> A diagrammatic tool that sets out sequential decisions, chance events, and payoffs in a tree-like structure to aid systematic analysis.
>
> **Key Term: expected value (EV)**
> The long-run average outcome of a decision, calculated as the sum of possible outcomes each weighted by its probability.
>
> **Key Term: conditional probability**
> The probability of one event occurring given that another event has already occurred.

## **Sequential Decisions and Conditional Probability**

Business decisions rarely involve only a single, isolated choice. Often, a manager must make decisions in stages: an initial decision is followed by uncertain events and subsequently by follow-up actions. In these cases, understanding sequential decisions and conditional probabilities is critical.

### Sequential Decisions

A sequential decision process involves several decisions and/or chance events that occur one after another. Each decision can affect the set of events that follow, as well as their probabilities and outcomes.

At each stage in a decision tree, branches represent possible choices or outcomes. By forecasting the outcomes and their probabilities at every point, you create a complete picture of all possible paths and consequences.

### Conditional Probability in Decision Trees

Conditional probability occurs when the probability of an event depends on the outcome of a previous event or decision. In a sequential decision tree, these dependencies need to be reflected in the calculation of expected values at each branch.

For example, the probability of success for a new product launch following a positive initial test market may be different from after a negative test market. Decision trees must use the correct conditional probabilities at each stage.

> **Key Term: rollback analysis**
> The process of working backwards through a decision tree, at each chance or decision node computing the value of the best available option, to identify the optimal decision path.

## **How to Construct and Analyse a Sequential Decision Tree**

1. **Define the decision problem.** Clarify the sequence of decisions and possible events, along with available information at each stage.
2. **Draw the tree structure.** Use squares for decision nodes (where you choose) and circles for chance nodes (where an uncertain event occurs).
3. **Assign probabilities to each chance event.** Use appropriate conditional probabilities where required.
4. **Record payoffs at the end of each possible path.** These represent financial outcomes, cost savings, or other relevant benefits.
5. **Perform rollback analysis.** Starting from the rightmost branches, calculate expected values at each chance node and compare payoffs at each decision node to select the best course of action.

### **Worked Example 1.1**

A pharmaceutical company must decide whether to conduct a small pilot trial (cost \$80,000) before a nationwide launch of a new drug. If successful, national launch yields profit of \$400,000; if not, the loss is \$100,000 (due to launch costs). If the pilot is skipped and the product is launched directly, the probability of success is 0.6; if the pilot is run, and successful, the probability of national launch success rises to 0.9. If the pilot fails, the launch is cancelled (no further profit or loss). The probability that the pilot is successful is 0.7.

**Draw the decision tree and, using rollback analysis, determine whether to run the pilot or proceed directly.**

> **Answer:**
>
> - **Option 1: Launch direct**: EV = 0.6 × \$400,000 + 0.4 × (−\$100,000) = \$240,000 − \$40,000 = \$200,000.
> - **Option 2: Run pilot**:
>   - Pilot cost: −\$80,000 (incurred regardless).
>   - Probability pilot succeeds: 0.7. Then, launch national:
>     - National launch success (0.9): \$400,000
>     - National launch failure (0.1): −\$100,000
>     - EV for national launch: 0.9 × \$400,000 + 0.1 × (−\$100,000) = \$360,000 − \$10,000 = \$350,000
>   - EV after pilot success: \$350,000
>   - Probability pilot fails: 0.3. No national launch. Profit/loss: \$0.
>   - Total EV: (0.7 × \$350,000) + (0.3 × \$0) = \$245,000
>   - Subtract pilot cost: \$245,000 − \$80,000 = \$165,000
>     Since direct launch EV (\$200,000) > pilot then launch EV (\$165,000), the company should launch directly without a pilot.

## **Rollback Analysis**

Rollback analysis is the method for evaluating decision trees with sequential stages. At each node, you calculate either the highest expected value (for a decision node) or the expected value itself (for a chance node), moving from the end branches back to the starting point. This ensures the optimal set of decisions is identified, factoring in all uncertainties at each stage.

**Procedure:**

1. Calculate expected value at each chance node (multiply payoffs by their probabilities, sum results).
2. At each decision node, select the branch with the highest expected value.
3. Move backwards, node by node, until the entire tree is resolved and an optimal starting decision is identified.

### **Worked Example 1.2**

A technology firm considers whether to develop advanced software at an R&D cost of \$50,000. After development, there is a 60% chance of high market demand (sales revenue \$300,000) and a 40% chance of low demand (\$40,000 revenue). Alternatively, they can first conduct market research for \$15,000. If research is positive (probability 0.5), the chance of high demand rises to 0.9. If research is negative (probability 0.5), the chance of high demand is 0.2. After negative research, they can still proceed or abandon.

**Should the company conduct market research before launching?** (Ignore discounting.)

> **Answer:**
>
> - **Without research**: EV = 0.6 × (\$300k−\$50k) + 0.4 × (\$40k−\$50k) = 0.6 × \$250k + 0.4 × (−\$10k) = \$150k − \$4k = \$146k
> - **With research**:
>   - Research cost: \$15k
>   - Positive research (0.5): Proceed with development.
>     - Success: 0.9 × (\$300k−\$50k) = 0.9 × \$250k = \$225k
>     - Failure: 0.1 × (\$40k−\$50k) = 0.1 × (−\$10k) = −\$1k
>     - EV = \$225k − \$1k = \$224k
>   - Negative research (0.5): Two options:
>     - Proceed:
>       - High demand: 0.2 × \$250k = \$50k
>       - Low demand: 0.8 × (−\$10k) = −\$8k
>       - EV = \$50k − \$8k = \$42k
>     - Abandon: EV = \$0
>     - Choose higher EV: \$42k
>   - Expected value after research: (0.5 × \$224k) + (0.5 × \$42k) = \$112k + \$21k = \$133k
>   - Subtract research cost: \$133k − \$15k = \$118k
>     Since \$146k (no research) > \$118k (do research), launch without research yields higher expected value.

### **Revision Tip**

Focus your practice on drawing full trees for multi-stage decisions and carefully assign probabilities at each branch, especially where prior results change the likelihood of later outcomes.

### **Exam Warning**

Be precise with conditional probabilities. Using simple unconditional probabilities at all stages is a common error—read the scenario to determine whether each branch's probability depends on earlier events.

## **Summary**

- Sequential business decisions often involve multiple stages, each influenced by previous choices and events.
- Conditional probabilities must be carefully assigned where outcomes depend on earlier branches.
- Decision trees provide a structured approach to compare alternative strategies and their risks.
- Rollback analysis is essential to solve trees efficiently and select the optimal path.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Define and calculate probability, decision tree payoffs, and expected value (EV)
- Apply conditional probabilities in sequential decisions
- Draw and analyse decision trees for multi-stage business decisions
- Use rollback analysis to identify optimal options in complex scenarios
- Recognise the significance of correct probability assignment in step-by-step decision making

## **Key Terms and Concepts**

- probability
- decision tree
- expected value (EV)
- conditional probability
- rollback analysis

Probability and decision trees - Sequential decisions and rollback