Practice Risk in Math

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

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Quick 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?

Showing a random 20 of 50 problems.

Example 1

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

Example 2

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 3

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 4

hard
A game pays $100 with probability pp and $0 otherwise. For what pp is a $15 ticket fairly priced?

Example 5

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 6

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 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

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 9

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 10

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 11

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 12

easy
If P(loss)=0.08P(\text{loss}) = 0.08 and the loss amount is $1{,}500, the expected loss is ___.

Example 13

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 14

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

Example 15

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 16

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 17

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 18

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

Example 19

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 20

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?
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