Decision analysis

From probabilities to decisions: expected utility

Predicting an outcome is not the same as choosing an action. Add preferences to your probability model and you can compare decisions on a consistent scale.

By the BayesLab team · Updated 18 September 2026 · 6 min read

Separate what you know from what you choose

A student club is choosing an outdoor or indoor event. Weather is uncertain, but Venue is a choice. Use a chance node Weather with states Dry/Wet and a decision node Venue with states Outdoor/Indoor. A utility node Satisfaction depends on both. This is a one-decision teaching exercise using fictional satisfaction points, not a financial forecast.

Let P(Dry) = 70% and P(Wet) = 30%. Outdoors scores 100 points if dry and −40 if wet. Indoors scores 60 if dry and 50 if wet. Utility expresses the preferences we have chosen for this exercise; it is not a probability and its rows do not sum to one.

Average each action over the uncertainty

Expected utility is a probability-weighted average of the utility for each possible outcome. Evaluate the same weather distribution for each action. Do not average the actions together: choosing between them is the purpose of the calculation.

EU(Outdoor) = 0.70 × 100 + 0.30 × (−40) = 58

EU(Indoor) = 0.70 × 60 + 0.30 × 50 = 57

Outdoor wins by one point under these assumptions. That narrow margin is worth noticing. It means the recommendation can be sensitive to a small change in the weather belief or preference values. “Highest expected utility” does not mean a guaranteed good outcome.

Build the influence diagram

1. Open BayesLab and start a blank model. Add the Weather chance node, label its states Dry/Wet, and enter 70%/30% in its Table tab.
2. Add the Venue decision node and name its choices Outdoor/Indoor.
3. Add a Satisfaction utility node. Connect Weather → Satisfaction and Venue → Satisfaction.
4. Enter the four utilities, carefully matching the displayed weather and venue labels. Enter −40 as a negative number, not a percentage.
5. Use the Decision panel to compare the fixed choices. The expected utilities should be 58 and 57.
6. Change Weather to 60% Dry / 40% Wet and compare again: Outdoor becomes 44, Indoor becomes 56.

Find the switching point

Let p be the probability of dry weather. Outdoor has utility 140p − 40 and Indoor has utility 10p + 50. They are equal when 130p = 90, so p = 9/13, approximately 69.23%. Above that threshold Outdoor is preferred; below it Indoor is preferred.

Distinguish revising a probability from observing weather. Changing the prior to 60% still leaves uncertainty. Setting Weather = Dry as evidence treats dry weather as known. Those experiments answer different questions and should not be mixed in your explanation.

Further reading: Berkeley CS188’s introduction to decision networks. Next, practise checking assumptions with the debugging walkthrough.