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.

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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: Decision under uncertainty means choosing the option with the best probability-weighted outcome, not the one with the flashiest single result.

Common stuck point: The procedure for decision under uncertainty is the easy part; the trap is choosing the option with the largest possible payoff. Asking "Am I choosing an action by weighing each option's outcomes against their probabilities?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I choosing an action by weighing each option's outcomes against their probabilities?

Worked Examples

Example 1

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

Answer

Bring umbrella (EV=-1) vs. don't bring (EV=-4). Bringing umbrella is optimal under expected value.

First step

1
Expected value of bringing umbrella: E(bring)=−1E(\text{bring}) = -1 (certain cost)

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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-\$20K). Calculate EV for each and identify which a risk-neutral investor and a risk-averse investor would choose.

Example 3

medium
A manufacturer can install quality control (cost $5000\$5000) or skip it. Without QC, defects occur with probability 0.150.15 and cost $50000\$50000. Compare expected costs.

Example 4

medium
A trader can hedge (cost $2k\$2k) protecting against a $30k\$30k loss that happens with p=0.1p=0.1. Compare expected costs.

Example 5

hard
Computing EVPI: without info you pick the EV-best act (EV =80= 80). Perfect info gives expected best-case payoff 100100. What is the EVPI and what does it represent?

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.

Example 3

easy
Option A: guaranteed $50. Option B: 50%50\% chance of $100, else $0. Compute B's expected value.

Example 4

easy
A lottery costs $2 and pays $10 with probability 0.10.1. Is the expected value positive or negative?

Example 5

easy
Two choices have equal expected value, but one can lose everything. Which does a risk-averse person prefer?

Example 6

easy
Why is choosing a stock 'because it could 10×10\times' a flawed decision rule?

Example 7

easy
An insurance policy has negative expected value for the buyer. Why might buying it still be rational?

Example 8

easy
Game X: EV =$5= \$5. Game Y: EV =$3= \$3. On expected value alone, which is better?

Example 9

easy
Ignoring a 0.1%0.1\% chance of a flood that would destroy your home — what decision error is this?

Example 10

easy
A decision tree has two branches: $30 at p=0.5p=0.5 and $10 at p=0.5p=0.5. What is the expected value?

Example 11

medium
A vendor offers: pay $20 for a 30%30\% chance to win $100. Compute the expected net value and decide whether to play on EV grounds.

Example 12

medium
Project A: $10k profit at p=0.9p=0.9, else lose $5k. Project B: sure $6k. Compute A's EV and choose on EV grounds.

Example 13

medium
Umbrella decision: rain probability 40%40\%. Carrying an umbrella costs 11 unit of hassle; getting soaked costs 1010. Compute expected cost of carrying vs not carrying.

Example 14

medium
Two investments have EV \$100 each. A: outcomes \$90 or \$110. B: outcomes \$0 or \$200. Which has higher risk, and how would a risk-neutral investor choose?

Example 15

medium
A startup bet: 5%5\% chance of $10M, 95%95\% chance of $0. Founder's alternative: a sure $200k salary over the same period. Compare EVs and discuss why a founder might still take the salary.

Example 16

medium
A factory must stock spare parts. Each spare costs $100; a stockout costs $1000 and occurs with probability 0.20.2 per period if unstocked. Is buying one spare justified on expected-cost grounds?

Example 17

medium
Maximin vs expected value: Option A worst case \$10, EV \$40. Option B worst case \$30, EV \$35. Which does a maximin (worst-case) decision-maker pick, and which does an EV-maximizer pick?

Example 18

medium
A warranty costs $40. Without it, a $300 repair is needed with probability 0.10.1. Compare expected costs.

Example 19

medium
Two routes to work: A averages 3030 min with low variance; B averages 2525 min but ranges 1515–6060 min. For an important meeting, which is the safer choice and why?

Example 20

challenge
A merchant can buy 00, 11, or 22 units of perishable stock at $4 each, selling at $10. Demand is 11 unit (p=0.5p=0.5) or 22 units (p=0.5p=0.5); unsold stock is worthless. Find the order quantity maximizing expected profit.

Example 21

challenge
Expected value of perfect information: without info you choose the higher-EV act (EV \$40). With perfect foresight you would always pick the best outcome, giving expected \$55. What is the EVPI, and what does it bound?

Example 22

challenge
St. Petersburg-style: a game pays $2^n where nn is the first toss that lands heads (probability 1/2n1/2^n). Show the expected payout diverges, and explain why people still pay only a small amount.

Example 23

easy
A fair coin flip pays $10\$10 on heads, $0\$0 on tails. What is the expected payout?

Example 24

easy
A bet costs $1\$1 and wins $5\$5 with probability 0.10.1, else $0\$0. Find the expected net value.

Example 25

easy
Outcomes: $20\$20 with p=0.5p=0.5, $80\$80 with p=0.5p=0.5. Find the expected value.

Example 26

easy
Option A: $100\$100 for sure. Option B: 50%50\% chance of $300\$300, 50%50\% chance of −$100-\$100. Which has higher expected value?

Example 27

easy
If a decision has expected value −$5-\$5, should a risk-neutral player play repeatedly?

Example 28

easy
A 1%1\% chance to win $1000\$1000 for $5\$5. What is the expected gain?

Example 29

medium
A drug treatment cures with probability 0.70.7 (no side effects) and harms with probability 0.30.3 (cost $1000\$1000). Placebo: no effect, no cost. EV of treatment vs placebo?

Example 30

medium
Two bets: A wins $10\$10 with p=0.6p=0.6; B wins $25\$25 with p=0.3p=0.3. Both cost the same. Higher EV?

Example 31

medium
A college applicant can apply to a safety school (admit p=1p=1, value 5050) or a reach school (admit p=0.2p=0.2, value 200200). EV comparison.

Example 32

medium
Outcomes for action A: $10\$10 (p=0.5p=0.5), −$2-\$2 (p=0.5p=0.5). Find EV and variance.

Example 33

medium
A vaccine costs $20\$20 and prevents a disease (p=0.05p=0.05 unvaccinated, cost $2000\$2000). Compute EV cost of vaccinating vs not.

Example 34

medium
A grocer can stock 00, 11, or 22 loaves of bread. Each costs $1\$1 and sells for $3\$3. Demand: 00 (p=0.2p=0.2), 11 (p=0.5p=0.5), 22 (p=0.3p=0.3). Find the EV-maximizing order.

Example 35

hard
A startup founder values $1 of certain salary as much as $2 of expected startup equity (due to risk aversion). Sure salary $200k\$200k vs equity worth $300k\$300k in expectation. Which does the founder pick?

Example 36

hard
Sequential decision: invest $100\$100 now for $400\$400 if a startup succeeds (p=0.4p=0.4), else $0\$0. Alternatively, wait one period to learn the outcome at cost $10\$10 (then choose). EV with vs without waiting?

Example 37

hard
A gambler doubles their bet after every loss (martingale). With a 50%50\% win chance and $1000\$1000 bankroll, why is this strategy ruinous?

Example 38

hard
A factory's expected downtime cost is \$50k/year. Preventive maintenance costs \$30k/year and reduces downtime cost to \$10k/year. EV comparison?

Example 39

hard
A medical test costs $200\$200 and lets you avoid surgery costing $10000\$10000 unnecessarily. Surgery is genuinely needed with probability 0.60.6. EV-justify the test (assume the test is perfectly informative).

Example 40

hard
A risk-neutral investor pays for an asset based on its expected payoff. If outcomes are $50\$50 (p=0.4p=0.4) and $100\$100 (p=0.6p=0.6), what is the fair price?

Example 41

challenge
Kelly criterion: with a bet that pays b:1b:1 on win probability pp, optimal fraction of wealth to risk is f∗=bp−(1−p)bf^* = \frac{bp - (1-p)}{b}. Compute f∗f^* for b=2,p=0.5b=2, p=0.5.

Example 42

challenge
Allais paradox-style preference: many people prefer (A) sure $1M\$1M over (B) 89%89\% sure $1M\$1M, 10%10\% chance $5M\$5M, 1%1\% chance $0\$0. Compare EVs.
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Background Knowledge

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

expected valueriskprobabilistic thinking