Expected Values and Risk

Expected value calculations and risk assessment questions test your ability to combine probability with numerical outcomes to make rational decisions. These questions are particularly relevant to the UCAT because they mirror the kind of reasoning clinicians use when evaluating treatment options, diagnostic tests, and public health interventions.

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What Is Expected Value?

The expected value (EV) is the average outcome you would expect if an event were repeated many times. It is calculated by multiplying each possible outcome by its probability and summing the results.

$EV = \sum (\text{outcome} \times \text{probability of that outcome})$EV=∑(outcome×probability of that outcome)

Simple Example

A game costs £2 to play. You roll a fair die. If you roll a 6, you win £10. Otherwise, you win nothing. Should you play?

Outcome	Probability	Value (net of cost)
Roll 6	1/6	£10 − £2 = £8
Roll 1–5	5/6	£0 − £2 = −£2

$EV = \frac{1}{6} \times 8 + \frac{5}{6} \times (-2) = \frac{8}{6} - \frac{10}{6} = -\frac{2}{6} \approx -£0.33$EV=61​×8+65​×(−2)=68​−610​=−62​≈−£0.33

The expected value is negative (−£0.33 per game), so on average you lose money. You should not play if making a rational economic decision.

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Expected Value in Medical-Style Scenarios

The UCAT frequently frames expected value questions in medical or health-related contexts.

Worked Example: Comparing Treatment Options

A hospital is choosing between two treatment protocols for a condition:

Treatment A: 70% chance of full recovery (value: 100), 20% chance of partial recovery (value: 60), 10% chance of no improvement (value: 0).

Treatment B: 50% chance of full recovery (value: 100), 50% chance of partial recovery (value: 60).

Treatment A EV: $EV_A = 0.70 \times 100 + 0.20 \times 60 + 0.10 \times 0 = 70 + 12 + 0 = 82$EVA​=0.70×100+0.20×60+0.10×0=70+12+0=82

Treatment B EV: $EV_B = 0.50 \times 100 + 0.50 \times 60 = 50 + 30 = 80$EVB​=0.50×100+0.50×60=50+30=80

Treatment A has a higher expected value (82 vs 80). Based purely on expected value, Treatment A is the better option.

Nuance: In the UCAT, you may need to consider more than just expected value. If Treatment A has a 10% risk of no improvement at all, a risk-averse patient might prefer Treatment B, which guarantees at least partial recovery. This kind of reasoning can appear in DM questions about assumptions and arguments.

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Risk Assessment

Risk assessment questions ask you to evaluate the probability and impact of negative outcomes. The key formula is:

$\text{Risk} = \text{Probability of harm} \times \text{Severity of harm}$Risk=Probability of harm×Severity of harm

Worked Example: Comparing Risks

Procedure X: 2% chance of a serious complication (severity score: 80) and 15% chance of a minor complication (severity score: 10).

Procedure Y: 5% chance of a moderate complication (severity score: 40) and 10% chance of a minor complication (severity score: 10).

Risk score for Procedure X: $R_X = 0.02 \times 80 + 0.15 \times 10 = 1.6 + 1.5 = 3.1$RX​=0.02×80+0.15×10=1.6+1.5=3.1

Risk score for Procedure Y: $R_Y = 0.05 \times 40 + 0.10 \times 10 = 2.0 + 1.0 = 3.0$RY​=0.05×40+0.10×10=2.0+1.0=3.0

Procedure Y has a marginally lower overall risk score (3.0 vs 3.1). However, the difference is small, and the nature of the risks differs — Procedure X has a small chance of a very serious event, while Procedure Y's worst outcome is moderate.

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Number Needed to Treat (NNT)