Practice Decision Under Uncertainty in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick 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.

Showing a random 20 of 50 problems.

Example 1

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 2

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 3

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 4

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 5

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 6

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 7

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 8

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 9

easy
A risk-averse person prefers a sure $50\$50 over a 50%50\% chance at $100\$100. Which decision rule did they apply?

Example 10

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 11

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

Example 12

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 13

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

Example 14

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 15

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?

Example 16

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 17

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 18

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.

Example 19

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

Example 20

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?
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