Decision Making using Probability: formula sheet

Key formulas

Classical probability

P(A) = number of favourable outcomes ÷ total number of equally likely outcomes

Use only when all outcomes are equally likely.

Empirical probability

P(A) = frequency of A ÷ total number of trials

Based on past data. It is an estimate, and improves with more observations.

Range and complement

0 ≤ P(A) ≤ 1; P(not A) = 1 − P(A)

Use the complement for 'at least one' type questions.

Sum of exhaustive, mutually exclusive outcomes

Σ P(outcomes) = 1

Always check that your probability distribution totals 1.

General addition rule

P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

Works for any two events.

Addition rule for mutually exclusive events

P(A ∪ B) = P(A) + P(B)

Valid only when P(A ∩ B) = 0.

General multiplication rule

P(A ∩ B) = P(A) × P(B | A)

Works for any two events with P(A) > 0.

Multiplication rule for independent events

P(A ∩ B) = P(A) × P(B)

Valid only when A and B are independent.

Conditional probability

P(B | A) = P(A ∩ B) ÷ P(A)

Defined for P(A) > 0.

Expected value

E(X) = Σ (Xᵢ × Pᵢ)

Xᵢ is each outcome and Pᵢ its probability. The probabilities must add up to 1.

Decision rule for profit or inflow

Choose the alternative with the highest EMV

Use for profit, contribution, revenue or cash inflow.

Decision rule for cost

Choose the alternative with the lowest expected cost

Use for cost or outflow problems.

Expected profit from a profit table

EMV of alternative = Σ (Profit under each event × Probability of that event)

Compute one EMV for each row (alternative).

Probability check

Σ P = 1

Check this first. If given probabilities do not add to 1, re-read the question.

Net EMV of a venture

Net EMV = EMV of inflows − Cost of the alternative

Deduct any upfront outlay that is specific to the alternative before comparing.

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.

Expected monetary value of an action

EMV = Σ (probability of state × payoff of the action in that state)

Take the highest EMV for profits and the lowest expected cost for costs. This is your decision without information.

Expected value with perfect information

EVwPI = Σ (probability of state × best payoff in that state)

Use the largest payoff in each state for profit, and the smallest cost in each state for cost problems.

EVPI for profit problems

EVPI = EVwPI − best EMV without information

Never negative. It is the maximum price worth paying for perfect information.

EVPI for cost problems

EVPI = lowest expected cost without information − expected cost with perfect information

The direction flips because lower cost is better.

Opportunity loss check

EVPI = minimum EOL, where opportunity loss = best payoff in the state − payoff of the action

The action with the best EMV also has the lowest EOL. Use this to verify your answer.

EVSI and net gain

EVSI = EV with sample information − EV without information; net gain = EVSI − cost of the information

EVSI ≤ EVPI. Buy the information only if the net gain is positive.

Maximax

Choose the action with the highest of the row maximum payoffs

Optimist's rule. If the table shows costs, choose the lowest of the row minimum costs (minimin).

Maximin

Choose the action with the highest of the row minimum payoffs

Pessimist's rule. If the table shows costs, choose the lowest of the row maximum costs (minimax cost).

Regret (opportunity loss)

Regret = best payoff in that state (column) − payoff of the action in that state

Build a regret table state by state. For costs, regret = actual cost − lowest cost in that state.

Minimax regret

Choose the action with the smallest of the row maximum regrets

Savage's criterion. Regret is never negative.

Laplace (equal likelihood)

Average payoff = sum of payoffs in the row ÷ number of states

Choose the highest average (lowest for costs).

Hurwicz criterion

Weighted value = α × row maximum + (1 − α) × row minimum

α is the coefficient of optimism, between 0 and 1. Choose the action with the highest value.

Expected value

EV = Σ (p × X)

Use when probabilities are known (risk). Probabilities must add up to 1.

Variance and standard deviation

σ² = Σ p × (X − EV)² and σ = √σ²

Equivalent shortcut: σ² = Σ p × X² − EV². Weights are the probabilities, not 1 ÷ n.

Coefficient of variation

CV = σ ÷ EV (× 100 for a percentage)

Relative risk per rupee of expected return. Lower CV means lower relative risk.

Quick revision

• Probabilities of all outcomes of an event must add up to 1.
• EMV of an action = Σ (payoff × probability of that outcome).
• Choose the highest EMV for profit, the lowest EMV for cost.
• EMV is an average over many repeats; a single decision may give a very different result.
• In a decision tree, squares are decisions and circles are chance events.
• Fold a tree back from right to left, using EMV at each chance node.
• Subtract costs on the branches that incur them, and show the net EMV of each option.
• EVPI = EMV with perfect information − best EMV without it.
• EVPI is the maximum you should pay for perfect information.
• Maximax picks the best of the best outcomes; maximin picks the best of the worst.
• Minimax regret picks the action with the smallest maximum regret.
• End every answer with a clear recommendation.

Common mistakes

• Adding probabilities of events that overlap without subtracting P(A ∩ B). Fix: Ask whether both can happen together. If yes, use P(A) + P(B) − P(A ∩ B).
• Treating mutually exclusive and independent as the same thing. Fix: Mutually exclusive means they cannot occur together. Independent means one does not affect the other. Two events with non-zero probabilities cannot be both.
• Using the simple average of outcomes instead of weighting by probability. Fix: Always write each outcome next to its probability and multiply before adding.
• Probabilities that do not add up to 1 are used without checking. Fix: Add the probabilities first. If the sum is not 1, re-read the question before calculating.
• Solving from left to right. Fix: Always start at the rightmost payoffs and move left. Later choices must be valued first.
• Forgetting to deduct the cost of an option. Fix: At each decision node compute EMV minus cost for every option before comparing.
• Picking the best payoff across the whole table instead of the best in each state. Fix: Work column by column. Perfect information means you choose the best action after knowing the state.
• Taking the highest value in a cost problem. Fix: Underline at the start whether you minimise or maximise. For costs use the lowest cost per state, and reverse the subtraction.
• Building the regret table row by row instead of column by column. Fix: Regret is measured against the best payoff in each state. Find the best figure in each column first, then subtract each payoff from it.
• Applying maximax or maximin directly to a cost table. Fix: Check whether the payoffs are profits or costs. For costs, the best outcome is the lowest figure, so the optimist's rule picks the lowest cost and the pessimist's rule picks the lowest of the worst (highest) costs.

Exam tips

• In Section A, read each option for the overlap trap: one option will usually be the sum without subtracting the intersection.
• State the independence or mutual exclusivity assumption in one line in written answers. It earns method marks.
• Check that a given probability distribution totals 1 before using it in an expected value working.
• For case-based MCQs, list events and probabilities in a small line-up first, then answer all linked questions from it.
• End decision questions with a one-line business meaning of your probability, not only the number.
• In the MCQ section, an EMV question is usually one table and one multiplication chain. Do it directly on the paper and check that probabilities add to 1.
• In written questions, show the table of outcomes and the EMV working for every alternative. Marks are given for method even if one figure slips.
• Read the wording: 'maximise profit' means highest EMV, 'minimise cost' means lowest EMV.