## **Learning Outcomes**

After reading this article, you will be able to apply probability concepts in decision making, construct and interpret decision trees, calculate expected values for different outcomes, and update probabilities using Bayes' theorem. You will understand how to present and use probabilistic information for risk analysis in performance management scenarios.

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

For ACCA Advanced Performance Management (APM), you are required to understand the application of probability in strategic decision-making. This article addresses key syllabus areas by focusing on:

- The role of probability and decision trees in risk and uncertainty analysis
- The calculation and significance of expected values in repeated and single-event decisions
- The practical use of Bayes' theorem to revise prior probabilities based on new information
- The ability to evaluate and improve decision-making processes using probabilistic models

## **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 is the expected value (EV) and how should it be used in performance management decisions?
2. How does a decision tree represent different choices and outcomes in uncertain scenarios?
3. Which management behaviour is most likely to choose the decision with the highest expected value?
   a) Risk averse  
   b) Risk seeking  
   c) Risk neutral  
   d) Uninformed
4. Briefly explain how Bayes' theorem helps decision makers in the context of uncertain outcomes.

## **Introduction**

Performance management regularly requires decisions under uncertainty. Probabilistic models help managers evaluate outcomes when not all events are certain. Using probability, expected values, and decision trees, management accountants can present objective analyses of likely results. Furthermore, updating probabilities as new information becomes available, by employing Bayes' theorem, allows better-informed decisions that maximize organisational objectives.

> **Key Term: probability**
> The likelihood of an event occurring, expressed as a value between 0 (impossible) and 1 (certain).

## Probability in Decision-Making

Uncertainty is inherent in most business environments. Probabilities quantify uncertainty, providing a logical basis for assigning likelihoods to different outcomes, such as market growth, competitor behaviour, or project success.

> **Key Term: decision tree**
> A diagram that maps out possible choices, chance events, probabilities, and corresponding outcomes to support structured decision-making.

## Expected Values

Expected value (EV) is a central tool for evaluating choices under uncertainty. It represents the long-run average outcome if a particular decision were repeated many times.

The calculation for EV is:

$$
EV = \sum (\text{probability} \times \text{outcome value})
$$

EVs allow managers to compare strategies based on their weighted average results. Risk-neutral decision makers prefer options with the highest expected value, regardless of individual outcome variance.

> **Key Term: expected value (EV)**
> The weighted average of all possible outcomes, using their probabilities as weights.

## Decision Trees in Practice

Decision trees visually structure complex decisions. Each branch represents a possible action or event. Choices branch as squares (decision points), while chance events branch as circles, with assigned probabilities for each scenario.

The process for using a decision tree:

1. Define decisions and chance events in sequence
2. Assign outcomes and their probabilities
3. Calculate expected values at each chance node
4. Work backwards (“rollback”) to identify the best strategy

### **Worked Example 1.1**

An energy company must decide between investing in a wind farm or solar panels. The profit from each depends on future regulation changes:

- For wind: 40% chance of extra subsidy (\$800k profit), 60% chance of base case (\$400k)
- For solar: 30% chance of high profit (\$900k), 70% chance of moderate profit (\$500k)

**Question:**  
Which investment has the higher expected value?

> **Answer:**
> Wind: (0.4 × \$800,000) + (0.6 × \$400,000) = \$320,000 + \$240,000 = \$560,000  
> Solar: (0.3 × \$900,000) + (0.7 × \$500,000) = \$270,000 + \$350,000 = \$620,000  
> Expected value for solar is higher (\$620,000), so a risk-neutral manager would prefer solar panels.

## Interpreting Expected Value and Risk

Expected value provides the basis for rational choice, but limitations exist:

- EV represents the average over many trials; it may not align with outcomes in a single occurrence decision.
- Variance (spread) and risk appetite must also be considered, especially for risk-averse or risk-seeking managers.

> **Key Term: risk appetite**
> The willingness of an organisation or manager to accept risk to achieve objectives.

## Revising Probabilities: Bayes' Theorem

Initial probability estimates (prior probabilities) are often based on historical data or judgement. New information can alter these beliefs. Bayes' theorem provides an objective way to revise probabilities—known as posterior probabilities—using observed evidence (such as market research or test results).

> **Key Term: Bayes' theorem**
> A mathematical rule for updating probabilities after considering new evidence.
>
> **Key Term: prior probability**
> The initial probability estimate for an event, before new data or evidence is considered.
>
> **Key Term: posterior probability**
> The revised probability of an event, after incorporating new evidence.

Bayes' theorem formula:

$$
\text{Posterior probability} = \frac{(\text{Probability of evidence given event}) \times (\text{Prior probability})}{\text{Total probability of evidence}}
$$

### **Worked Example 1.2**

A company estimates a 60% chance a new product will succeed in the market (prior probability). It commissions market research, which is 80% reliable in detecting successful products. The research predicts success.

**Question:**  
What is the revised probability (posterior) that the product will succeed, given a positive research result?  
Assume that the test predicts success 80% of the time when the product truly will succeed, and 25% of the time when it will fail.

> **Answer:**
>
> - Probability product will succeed and test predicts success: 0.6 × 0.80 = 0.48
> - Probability product will fail and test predicts success: 0.4 × 0.25 = 0.10
> - Total probability test predicts success: 0.48 + 0.10 = 0.58
> - Posterior probability (product will succeed | test predicts success): 0.48 / 0.58 ≈ 0.83 (83%)

## Applications and Limitations in ACCA APM

- Use EV to compare long-term projects or decisions occurring frequently.
- Always check whether probabilities are reliable and relevant for the scenario.
- Use Bayes’ theorem when new, relevant information becomes available.
- In a one-off (single occurrence) decision, consider if EV is meaningful or whether management’s risk attitude favours an alternative approach.

### **Exam Warning**

> Probabilities in case studies are often based on assumptions. Justify your probability assessments, and be ready to update them using Bayes’ theorem if new evidence is given. Clearly state any assumptions made, as this may earn marks in the APM exam.

### **Revision Tip**

_Adding probabilities to your decision trees and showing rollback calculations step-by-step can score easy marks. Always label probability branches and outcomes clearly for the examiner._

## **Summary**

Probability and decision trees structure decisions under risk, allowing the calculation of expected values for different strategies. Bayes’ theorem offers a systematic way to revise probability assessments as new information arises, helping managers to make more accurate decisions in uncertain environments.

## **Key Point Checklist**

_This article has covered the following key knowledge points:_

- Use of probability to quantify uncertainty in management decisions
- Structure and interpretation of decision trees
- Calculation and practical meaning of expected value (EV)
- Limitations of EV and impact of risk appetite
- Revision of probabilities as new data becomes available using Bayes’ theorem
- Definitions of prior and posterior probabilities

## **Key Terms and Concepts**

- probability
- decision tree
- expected value (EV)
- risk appetite
- Bayes' theorem
- prior probability
- posterior probability

Probability and decision trees - Expected values and revision with bayes