Risk Examples in Math

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

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

Concept Recap

Risk is the possibility of loss or negative outcome, quantified by combining the probability of the event with the severity of its impact: Expected Loss = P(loss) times amount of loss.

What could go wrong, how likely is it, and how bad would it be?

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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: Risk weighs the probability of a loss against the size of that loss: Expected Loss =P(loss)×=P(\text{loss})\times amount of loss.

Common stuck point: The procedure for risk is the easy part; the trap is ranking by probability alone. Asking "Am I combining the probability of a loss with the size of that loss?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I combining the probability of a loss with the size of that loss?

Worked Examples

Example 1

medium
A business faces two possible disasters: (A) equipment failure — probability 0.10, cost \$50,000; (B) data breach — probability 0.02, cost \$500,000. Calculate the expected loss from each and determine which poses greater financial risk.

Answer

Expected loss: Equipment = \$5,000; Data breach = \$10,000. Data breach is the greater financial risk.

First step

1
Expected loss A: EA=P(A)×cost=0.10×50000=$5,000E_A = P(A) \times \text{cost} = 0.10 \times 50000 = \$5,000

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Example 2

hard
Should you buy insurance for \$200/year that covers a \$5000 loss occurring with probability 0.03? Calculate expected value for both choices and discuss why someone might still buy the insurance.

Example 3

medium
A bike costs $600. Theft insurance is $45/year. Last year 3%3\% of similar bikes were stolen. Compute the expected loss without insurance and decide whether the premium beats the expected loss.

Example 4

medium
A driver's annual probability of an accident is 0.050.05, with average repair cost $3{,}000. Compute her expected annual repair loss. What premium would let an insurer break even (ignoring overhead)?

Example 5

medium
You can pay $30 to insure a $600 phone with a 7%7\% chance of damage this year. Compute expected cost with and without insurance. Which is lower in expectation?

Example 6

medium
A startup has a 30%30\% chance of $1{,}000{,}000 in profit and a 70%70\% chance of breaking even. Find the expected profit and explain why investors might still demand a higher return than this.

Example 7

hard
Two independent risks A (P=0.10P=0.10, loss $1{,}000) and B (P=0.20P=0.20, loss $500) can both occur. Find the expected total loss and the probability that AT LEAST one occurs.

Example 8

hard
Plan A: certain $400 cost. Plan B: 20%20\% chance of $0 cost, 80%80\% chance of $500 cost. Compare expected cost and explain when a risk-averse person might still pick A.

Example 9

challenge
A bridge has a 0.1%0.1\% annual chance of collapse causing $50{,}000{,}000 in damage. The owner can spend $30{,}000 per year on maintenance that cuts the collapse probability in half. Is the maintenance worth it in expected value? By how much?

Practice Problems

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

Example 1

easy
A lottery ticket costs $2 and pays $1,000,000 with probability 12,000,000\frac{1}{2,000,000}. Calculate the expected value and determine whether this is a good financial decision.

Example 2

hard
Two investment options: A = certain gain of \$500; B = 60% chance of \$1000, 40% chance of \$0. Calculate EV for both. Which would a risk-neutral person choose? A risk-averse person?

Example 3

easy
An event has probability 0.20.2 of a $500\$500 loss. What is the expected loss?

Example 4

easy
Risk combines two things: the probability of a bad outcome and its ___.

Example 5

easy
Event A: 50% chance of losing \$10. Event B: 1% chance of losing \$10,000. Which has the higher expected loss?

Example 6

easy
True or false: a good decision can always achieve zero risk.

Example 7

easy
People fear plane crashes more than car accidents though cars kill far more. This gap is between perceived risk and ___ risk.

Example 8

easy
A lottery ticket has a 0.0010.001 chance to win and $0 otherwise; the downside is the $2 cost. What is the expected loss from the ticket price alone?

Example 9

easy
Which is the riskier investment: one with possible loss but unknown probability, or one with a known, very small loss probability and small amount?

Example 10

easy
An insurer charges a premium slightly above the expected loss per policy. What is this margin meant to cover?

Example 11

medium
A project has a 10% chance of a \$50,000 loss and a 90% chance of a \$5,000 gain. What is the expected monetary value?

Example 12

medium
Two options: A loses $20 with probability 0.30.3; B loses $5 with probability 0.90.9. Which has lower expected loss?

Example 13

medium
A factory faces a 0.5% annual chance of a \$2,000,000 fire loss. What annual expected loss should it budget?

Example 14

medium
Why might a rational person buy insurance even though the premium exceeds the expected loss?

Example 15

medium
Risk X: probability 0.0010.001, loss $1,000,000. Risk Y: probability 0.50.5, loss $100. Which has higher expected loss, and which feels scarier?

Example 16

medium
A bet wins $90 with probability 0.50.5 and loses $100 with probability 0.50.5. Is the expected value positive or negative?

Example 17

medium
A portfolio's loss probability is unknown but bounded between 1% and 5%, with loss \$200,000. Give the range of expected loss.

Example 18

medium
A warranty pays $300 if a device fails, which happens with probability 0.040.04. What expected payout per device should the seller price in?

Example 19

medium
Action A risks a $100 loss with probability 0.20.2; Action B risks a $40 loss with probability 0.60.6. Which has the lower expected loss?

Example 20

challenge
Insurer sells 10,000 policies; each has a 0.020.02 chance of a $5,000 claim. What total expected payout should it budget, and what premium per policy just covers expected loss?

Example 21

challenge
Compare: Plan A loses $1000 with p=0.1p=0.1; Plan B loses $300 with p=0.4p=0.4. Which to choose to minimize expected loss, and by how much do they differ?

Example 22

challenge
A decision has outcomes: gain $1000 (p=0.6p=0.6), lose $500 (p=0.3p=0.3), lose $2000 (p=0.1p=0.1). Find the expected value and state if it is worth taking on EV alone.

Example 23

easy
A storm has probability 0.250.25 of causing $2{,}000 in damage. What is the expected loss?

Example 24

easy
Which has higher expected loss: A) 30%30\% chance of losing $100, or B) 5%5\% chance of losing $1{,}000?

Example 25

medium
A factory faces two independent risks: fire (P=0.01P=0.01, loss $200{,}000) and flood (P=0.04P=0.04, loss $50{,}000). What is the total expected annual loss?

Example 26

medium
A $1 raffle ticket has a 1200\frac{1}{200} chance of winning $150. What is the expected net gain per ticket?

Example 27

medium
Stock A has a 40%40\% chance of dropping $500. Stock B has a 10%10\% chance of dropping $2{,}500. Which has the larger expected loss, and by how much?

Example 28

medium
A medical test has a 2%2\% chance of a serious side effect. If the side effect occurs, the cost is $8{,}000. The test is given to 5{,}000 patients. Estimate the total expected cost.

Example 29

medium
Project X: 70\% chance of \$10{,}000 profit, 30\% chance of \$4{,}000 loss. Compute the expected profit.

Example 30

medium
A delivery company estimates a 1%1\% daily chance of a $25{,}000 truck accident. What expected loss should it budget for a 365-day year?

Example 31

hard
A coin will be flipped 3 times. You lose \$30 if all three are heads. Compute the expected loss.

Example 32

hard
You can pay $X up front to avoid a 15%15\% chance of a $2{,}400 loss. For what $X is the deal exactly fair (no risk premium)?

Example 33

hard
A roofer offers a 1-year guarantee. He estimates a 4%4\% chance a repair will cost him $2{,}500 and a 1%1\% chance a major redo costs $10{,}000. How much should he add to each job to cover expected warranty losses?

Example 34

hard
An insurer sells 10{,}000 policies. Each has a 2%2\% chance of a $5{,}000 claim. The premium is $120 per policy. What is the insurer's expected annual profit?

Example 35

hard
A car has a 0.5%0.5\% chance of theft per year. Comprehensive insurance costs $300. For what car value VV is the premium fair?
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Background Knowledge

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

probability