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