Applying Expected Value in Multi-Step and Real-World Problems Quiz

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| Questions: 20 | Updated: Nov 16, 2025
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1) A carnival game costs $4 to play. You draw a ball with probabilities: 50% win $2, 30% win $4, 15% win $6, 5% win $20. What is the expected net gain/loss per play?

Explanation

Expected payout = 0.5·2 + 0.3·4 + 0.15·6 + 0.05·20 = 4.10; Net = 4.10 − 4 = +0.10 (gain).

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About This Quiz
Applying Expected Value In Multi-step And Real-world Problems Quiz - Quiz

This final quiz delves into multi-step problems where expected value calculations are used in complex scenarios like product launches, public transportation, and game decisions. It challenges you to calculate expected values in varied contexts, including business, insurance, and customer behavior analysis, while also exploring how time and costs influence decisions... see morebased on expected value. see less

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2) A stock has annual outcomes: 20% +30%, 50% +8%, 20% −5%, 10% −20%. On a $10,000 investment, what is the expected change in dollars after one year (simple expectation on returns)?

Explanation

Expected change = 0.2·3000 + 0.5·800 + 0.2·(−500) + 0.1·(−2000) = 600 + 400 − 100 − 200 = $700.

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3) A student plays a quiz game: each correct answer earns $3, each wrong answer loses $1.50. She answers correctly with probability 0.65 independently over 10 questions. What is the expected total gain?

Explanation

Expected per question = 0.65·3 + 0.35·(−1.5) = 1.425; For 10 questions: 10·1.425 = $14.25.

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4) A store offers a loyalty card: each purchase independently has a 10% chance to award a $5 coupon, otherwise $0. If you make 25 purchases, what is the expected total coupon value?

Explanation

Expected per purchase = 0.1·5 = 0.5; For 25 purchases: 25·0.5 = $12.50.

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5) A slot machine pays: $50 with prob 0.01, $10 with prob 0.05, $2 with prob 0.20, $0 otherwise. Cost per play is $2. What is the expected value per play?

Explanation

Expected payout = 0.01·50 + 0.05·10 + 0.2·2 = 0.5 + 0.5 + 0.4 = 1.4; Net = 1.4 − 2 = −$0.60.

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6) What is the expected net profit from Route A alone? (Route A revenue: $200 (30%), $260 (50%), $340 (20%), cost $150.)

Explanation

E[revenue A] = 0.3·200 + 0.5·260 + 0.2·340 = 258; Net = 258 − 150 = $108.

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7) What is the expected net profit from Route B alone? (Route B revenue: $240 (40%), $300 (40%), $420 (20%), cost $180.)

Explanation

E[revenue B] = 0.4·240 + 0.4·300 + 0.2·420 = 300; Net = 300 − 180 = $120.

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8) If both routes are run in a day, what is the expected total net profit including the bonus? (Bonus: +$25 with probability 0.6 if both routes are run.)

Explanation

Net routes = 108 + 120 = 228; Expected bonus = 0.6·25 = 15; Total = 228 + 15 = $243.

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9) If the company operates 5 days per week, what is the expected weekly net profit using both routes daily?

Explanation

Daily profit = 243; Weekly = 5·243 = $1,215.

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10) A raffle ticket costs $8. Prizes: one $500, four $100, ten $20. There are 1,000 tickets. What is the expected net gain/loss for buying one ticket?

Explanation

Expected prize = (1/1000)·500 + (4/1000)·100 + (10/1000)·20 = 0.5 + 0.4 + 0.2 = 1.1; Net = 1.1 − 8 = −$6.90.

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11) Draw one card from a standard 52-card deck. You win $12 for a heart, $5 for any face card (J/Q/K) that is not a heart, and lose $3 otherwise. What is the expected value?

Explanation

Hearts: 13 cards → (13/52)·12 = 3; Non-heart faces: 9 cards → (9/52)·5 ≈ 0.865; Others: 30 cards → (30/52)·(−3) ≈ −1.731; Total ≈ 3 + 0.865 − 1.731 = $2.13.

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12) Two independent machines produce widgets with defect rates 3% and 5% respectively. Each defective widget costs $15 to rework. If they produce 400 and 300 widgets respectively per day, what is the expected daily rework cost?

Explanation

Machine 1: 0.03·400 = 12 defects → 12·15 = 180; Machine 2: 0.05·300 = 15 defects → 15·15 = 225; Total = 180 + 225 = $405.

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13) A free-throw shooter hits with probability 0.78. In a game he expects 9 attempts. Each make is worth 1 point. If he is fouled on a made 3-pointer (prob 0.1 each attempt, independent) he gets one extra free throw worth the same EV per attempt. What is his expected total points from free throws?

Explanation

Base points: 9·0.78 = 7.02; Expected extra attempts: 9·0.1 = 0.9, each worth 0.78; Extra points: 0.9·0.78 = 0.702; Total = 7.02 + 0.702 = 7.722.

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14) An insurance policy charges $600. Claims: $0 with 92%, $3,000 with 6%, $10,000 with 2%. What is the insurer’s expected profit per policy?

Explanation

Expected claim = 0.06·3000 + 0.02·10000 = 180 + 200 = 380; Profit = 600 − 380 = $220.

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15) A coin with P(Heads)=0.55 is flipped until the first Heads or up to 3 flips, whichever comes first. You win $10 if you get Heads by the 3rd flip; otherwise lose $3. What is the expected value?

Explanation

P(no Heads in 3 flips) = (0.45)³ ≈ 0.0911; P(win) = 1 − 0.0911 ≈ 0.9089; EV = 10·0.9089 + (−3)·0.0911 ≈ 9.0889 − 0.2733 ≈ $8.82.

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16) What is the expected profit from launching only X (after overhead)? (Product X: 120k (40%), 50k (35%), −30k (25%). Overhead: 40k once per day.)

Explanation

E(X) = 0.4·120 + 0.35·50 + 0.25·(−30) = 58k; After overhead: 58 − 40 = 18k.

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17) What is the expected profit from launching only Y (after overhead)? (Product Y: 200k (25%), 80k (50%), −60k (25%). Overhead: 40k once per day.)

Explanation

E(Y) = 0.25·200 + 0.5·80 + 0.25·(−60) = 75k; After overhead: 75 − 40 = 35k.

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18) If they launch both X and Y on the same day (one overhead charge), what is the expected profit?

Explanation

E(X) + E(Y) = 58k + 75k = 133k; After one overhead: 133 − 40 = 93k.

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19) A commuter chooses between Train and Bus. Train: 25 min with 70%, 40 min with 30%. Bus: 30 min with 60%, 35 min with 40%. If time is valued at $18/hour, what is the expected monetary cost difference (Train − Bus) from time spent?

Explanation

E[Train] = 0.7·25 + 0.3·40 = 29.5 min; E[Bus] = 0.6·30 + 0.4·35 = 32 min; Difference = −2.5 min = −2.5/60 h ≈ −0.0417 h; Cost difference = −0.0417·18 ≈ −$0.75 (Train cheaper).

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20) A game: roll two fair dice. Win $15 if doubles, $8 if sum is 7, lose $3 otherwise. What is the expected value per play?

Explanation

P(doubles) = 6/36 = 1/6; P(sum 7) = 6/36 = 1/6; P(other) = 24/36 = 2/3; EV = 15·(1/6) + 8·(1/6) + (−3)·(2/3) = 2.5 + 1.333… − 2 = $1.83 (approx).

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A carnival game costs $4 to play. You draw a ball with probabilities:...
A stock has annual outcomes: 20% +30%, 50% +8%, 20% −5%, 10% −20%....
A student plays a quiz game: each correct answer earns $3, each wrong...
A store offers a loyalty card: each purchase independently has a 10%...
A slot machine pays: $50 with prob 0.01, $10 with prob 0.05, $2 with...
What is the expected net profit from Route A alone? (Route A revenue:...
What is the expected net profit from Route B alone? (Route B revenue:...
If both routes are run in a day, what is the expected total net profit...
If the company operates 5 days per week, what is the expected weekly...
A raffle ticket costs $8. Prizes: one $500, four $100, ten $20. There...
Draw one card from a standard 52-card deck. You win $12 for a heart,...
Two independent machines produce widgets with defect rates 3% and 5%...
A free-throw shooter hits with probability 0.78. In a game he expects...
An insurance policy charges $600. Claims: $0 with 92%, $3,000 with 6%,...
A coin with P(Heads)=0.55 is flipped until the first Heads or up to 3...
What is the expected profit from launching only X (after overhead)?...
What is the expected profit from launching only Y (after overhead)?...
If they launch both X and Y on the same day (one overhead charge),...
A commuter chooses between Train and Bus. Train: 25 min with 70%, 40...
A game: roll two fair dice. Win $15 if doubles, $8 if sum is 7, lose...
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