Expected Value & Risk Assessment — Practice

Practice bank for UCAT Decision Making questions involving expected values, risk comparisons, and decision-making under uncertainty.

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Quick-Reference: Expected Value

Formula

$E(X) = \sum [\text{value of outcome} \times \text{probability of outcome}]$E(X)=∑[value of outcome×probability of outcome]

The expected value is the weighted average of all possible outcomes.

Example

A treatment has a 70% chance of full recovery (value = 10) and a 30% chance of partial recovery (value = 4).

$E = (0.70 \times 10) + (0.30 \times 4) = 7.0 + 1.2 = 8.2$E=(0.70×10)+(0.30×4)=7.0+1.2=8.2

Comparing Options

When comparing two options, calculate the expected value of each and compare. The option with the higher expected value is generally preferable (all else being equal).

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Quick-Reference: Risk Measures

Measure	Formula	Meaning
Absolute risk	Events / Total	Probability of event occurring
Relative risk (RR)	Risk in exposed / Risk in unexposed	How much more likely the event is in the exposed group
Absolute risk reduction (ARR)	Risk in control − Risk in treatment	Actual reduction in risk
Relative risk reduction (RRR)	ARR / Risk in control	Proportional reduction in risk
Number needed to treat (NNT)	1 / ARR	How many patients must be treated to prevent one event

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Common Pitfalls

• Confusing relative risk reduction with absolute risk reduction (a 50% RRR sounds impressive, but if the baseline risk is 2%, the ARR is only 1%)
• Forgetting to convert percentages to decimals before calculating
• Comparing expected values without considering variance/risk tolerance

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Strategy

1. Identify all possible outcomes and their probabilities
2. Check that probabilities sum to 1 (or 100%)
3. Multiply each outcome value by its probability
4. Sum the products
5. Target: 45–60 seconds

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Practice

Complete the 10 assessment questions.