11.1 Risk, Uncertainty and Probability: Rules and Joint Probabilities for Decisions

Why this topic is examined

Syllabus area D1 asks you to demonstrate the impact of risk. Component D1(a) requires you to explain the concepts of risk and uncertainty, and D1(b) to use expected values and joint probabilities in decision-making, including "probability and its relationship with proportions and percentages". The syllabus also notes that candidates will not be asked to apply techniques to deal with uncertainty, so you need to understand the concept of uncertainty, but calculation questions focus on risk: probabilities, joint probabilities and expected values (Section 11.2).

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Risk and uncertainty

In practice the line is blurred: managers often attach subjective probabilities to uncertain situations. For the exam, remember the test: can probabilities be estimated?

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Probability basics

A probability measures how likely an event is, on a scale from 0 (impossible) to 1 (certain).

$$P(\text{event}) = \frac{\text{Number of ways the event can occur}}{\text{Total number of equally likely outcomes}}$$

Probability, proportions and percentages

Probabilities, proportions and percentages are three ways of expressing the same idea:

When probabilities come from past records they are called relative frequencies. Using them for the future assumes that conditions will stay similar.

Complementary events

The probability that an event does not happen is:

$$P(\text{not } A) = 1 - P(A)$$

If 12.5% of components are defective, the probability that a component is acceptable is 1 − 0.125 = 0.875.

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The addition rule ("or")

Mutually exclusive events cannot happen at the same time (a customer's order is either delivered on time or late, not both):

$$P(A \text{ or } B) = P(A) + P(B)$$

If events can happen together, the overlap must not be counted twice:

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$

Example: of 400 invoices checked, 60 contain a pricing error, 40 contain a quantity error and 20 contain both. The probability that a randomly chosen invoice contains at least one error is:

$$\frac{60}{400} + \frac{40}{400} - \frac{20}{400} = \frac{80}{400} = 0.20$$

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The multiplication rule ("and")

Independent events do not affect each other's probabilities:

$$P(A \text{ and } B) = P(A) \times P(B)$$

Dependent events do affect each other. Use the conditional probability of B given that A has occurred:

$$P(A \text{ and } B) = P(A) \times P(B \mid A)$$

Example (dependent): a box of 10 parts contains 2 defective parts. Two parts are drawn without replacement. The probability that both are defective is:

$$\frac{2}{10} \times \frac{1}{9} = \frac{2}{90} \approx 0.022$$

The second probability is 1/9 because one defective part has already been removed.

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Joint probabilities in decision-making

Many business problems involve two uncertain variables at once. A joint probability is the probability that a particular combination of outcomes occurs.

Worked example: sales volume and contribution per unit

A company is budgeting for a new product. It estimates:

The two variables are independent. Fixed costs are $41,000.

Joint probability table

From the table:

• The probability of making a profit is 0.42: only the combination of high volume and high contribution covers the $41,000 of fixed costs.
• The probability of making a loss is 0.18 + 0.28 + 0.12 = 0.58.
• The probability that total contribution is at least $40,000 is 0.42 + 0.18 + 0.28 = 0.88.

A probability tree shows the same information as branches: the first set of branches for volume, the second for contribution, with the joint probability at the end of each path found by multiplying along it.

Why joint probabilities matter

A single "most likely" budget (10,000 units at $5) shows a profit of $9,000, but the joint probability table reveals that a loss is more likely than not (58%). Probability analysis gives management a far better picture of risk than one point estimate. Section 11.2 summarises such tables with expected values, and Sections 11.3 and 11.4 measure spread with the standard deviation and the normal distribution.