Decision Under Uncertainty Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Decision Under Uncertainty.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Decision under uncertainty involves choosing between options whose outcomes are not known for certain, typically by comparing expected values or risk profiles.

The rational strategy under uncertainty is not always to pick the option with the best single outcome but the one with the best expected outcome weighted by its probability.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Combine probability of outcomes with their values (expected value) to decide.

Common stuck point: Expected value isn't everything—risk tolerance and worst-case scenarios matter too.

Worked Examples

Example 1

medium
An umbrella decision: if you bring an umbrella, you carry it all day (cost: mild inconvenience, -1). If you don't bring it and it rains (prob 0.4), you get wet (cost: -10). If you don't bring it and it doesn't rain (prob 0.6), cost is 0. Calculate expected values for both decisions.

Solution

  1. 1
    Expected value of bringing umbrella: E(\text{bring}) = -1 (certain cost)
  2. 2
    Expected value of not bringing: E(\text{no umbrella}) = 0.4(-10) + 0.6(0) = -4
  3. 3
    Optimal decision: bring umbrella (-1 > -4)
  4. 4
    Interpretation: expected cost is 4 times worse without umbrella — bring it

Answer

Bring umbrella (EV=-1) vs. don't bring (EV=-4). Bringing umbrella is optimal under expected value.
Decision under uncertainty uses expected value to compare options with probabilistic outcomes. Each option is evaluated by its probability-weighted average outcome. The action with the highest (or least negative) expected value is optimal for a risk-neutral decision-maker.

Example 2

hard
A startup has three investment options: A (safe: gain \50K certain), B (medium: 70% chance \80K, 30% chance \0), C (risky: 30% chance \200K, 70% chance -\20K$). Calculate EV for each and identify which a risk-neutral investor and a risk-averse investor would choose.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
You can either (A) study for certain +10 points on the final, or (B) not study: 50% chance +20 points, 50% chance +0 points. Calculate EV for both and decide.

Example 2

hard
A city decides whether to build a flood barrier (cost \10M). Flood probability in 50 years: 0.30; flood damage if no barrier: \50M; damage with barrier: \$5M. Calculate expected costs for building vs. not building.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

expected valueriskprobabilistic thinking