Multi-Step EV Applications Quiz (HSS.MD.A.1-2)

1) 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?

$2.13

$0.25

−$0.02

−$0.10

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.

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.

2) 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?

$0.24

$0.36

$0.00

$1.83

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).

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).

3) 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?

$−0.75

$−1.20

$0.00

$1.20

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).

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).

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

$67k

$93k

$73.5k

$113.5k

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

Explanation

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

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

$35k

$45k

$47.5k

$50k

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

Explanation

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

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

$32k

$23.5k

$18k

$12k

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

Explanation

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

7) 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?

$3.40

$4.00

$8.82

$9.00

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.

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.

8) 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?

$60

$120

$220

$−60

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

Explanation

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

9) 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?

7.02

7.722

7.8

8.502

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.

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.

10) 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?

$255

$405

$270

$285

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

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.

11) 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?

Gain of $0.30

Gain of $0.10

Loss of $0.10

Loss of $0.40

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

Explanation

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

12) 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?

−$6.20

−$5.80

−$6.90

−$4.80

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.

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.

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

$900

$1,100

$1,215

$1,250

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

Explanation

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

14) 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.)

$243

$228

$240

$250

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

Explanation

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

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

$84

$96

$108

$120

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

Explanation

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

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

$80

$95

$110

$108

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

Explanation

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

17) 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?

−$0.20

−$0.40

$0.00

−$0.60

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.

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.

18) 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?

$12.50

$7.50

$25.00

$50.00

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

Explanation

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

19) 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?

$7.50

$9.00

$12.00

$14.25

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

Explanation

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

20) 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)?

$600

$700

$740

$800

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

Explanation

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

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

1) 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?

2)

What first name or nickname would you like us to use?

2) 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?

3) 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?

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

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

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

7) 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?

8) 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?

9) 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?

10) 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?

11) 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?

12) 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?

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

14) 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.)

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

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

17) 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?

18) 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?

19) 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?

20) 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)?