← Back to the Expected Value Decisions worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: S-MD.A.3, S-MD.A.4, S-MD.B.5 · Level: Grade 12 Precalculus
Problem 1 Valid Distributions
(a) 0.2 + 0.3 + 0.4 + 0.2 = 1.1, which is more than 1, so it is not valid.
(b) 0.25 + 0.25 + 0.30 + 0.20 = 1.00 and every value is between 0 and 1, so it is valid.
Answer: (a) not valid; (b) valid
Problem 2 Missing Probability
The probabilities must add to 1: 0.15 + 0.35 + 0.20 = 0.70, so P(2) = 1 − 0.70 = 0.30.
Answer: 0.30
Problem 3 A Labeled Die
P(0) = 26 = 13, P(1) = 36 = 12, P(4) = 16.
E(X) = 0(13) + 1(12) + 4(16) = 36 + 46 = 76 ≈ 1.17.
Answer: E(X) = 76 ≈ 1.17
Problem 4 Expected Value from a Table
E(X) = 2(0.4) + 4(0.3) + 6(0.2) + 8(0.1) = 0.8 + 1.2 + 1.2 + 0.8 = 4.0.
Answer: 4.0
Problem 5 Library Survey
Divide by 30: P(0) = 0.2, P(1) = 0.4, P(2) = 0.3, P(3) = 0.1.
E = 0(0.2) + 1(0.4) + 2(0.3) + 3(0.1) = 0.4 + 0.6 + 0.3 = 1.3.
Answer: 1.3 books
Problem 6 Is It Fair?
E(winnings) = 12(14) + 0(34) = $3. Expected net gain = 3 − 3 = $0.
Answer: yes, the game is fair (expected net gain $0)
Problem 7 School Raffle
Total prizes: 300 + 2(50) = $400. Expected winnings per ticket = 400200 = $2.
Expected net gain = 2 − 5 = −$3.
Answer: $2.00 expected winnings; −$3.00 expected net gain
Problem 8 Ten-Sector Spinner
P($20) = 0.1, P($5) = 0.2, P($1) = 0.3, P($0) = 0.4.
E(payout) = 20(0.1) + 5(0.2) + 1(0.3) + 0 = 2 + 1 + 0.3 = $3.30. Expected net gain = 3.30 − 3.00 = $0.30.
Answer: expected net gain +$0.30 per spin
Problem 9 Free Throws
P(0) = 0.3 × 0.3 = 0.09. P(2) = 0.7 × 0.7 = 0.49. P(1) = 2(0.7)(0.3) = 0.42. The three add to 1.
E(X) = 0(0.09) + 1(0.42) + 2(0.49) = 0.42 + 0.98 = 1.40.
Answer: 0.09, 0.42, 0.49; E(X) = 1.4 shots
Problem 10 Insurance Policy
Expected loss = 0.03(2,500) = $75.
The policy costs $120 but covers an expected $75, so the expected net result of buying is 75 − 120 = −$45.
Answer: expected loss $75; buying the policy costs $45 more on average
Problem 11 Speaker Warranty
Expected repair cost = 0.12(180) = $21.60.
The warranty costs 55 − 21.60 = $33.40 more than the expected repair cost, so on average it is not worth buying.
Answer: expected repair cost $21.60 versus $55 warranty
Problem 12 Comparing Two Games
Game A: 5(0.3) − 2 = 1.50 − 2 = −$0.50. Game B: 3(0.6) − 2 = 1.80 − 2 = −$0.20.
Both lose money on average, but Game B loses less per play.
Answer: A: −$0.50; B: −$0.20; Game B is better
Problem 13 Televisions per Household
Relative frequencies: 0.05, 0.25, 0.45, 0.20, 0.05 (counts 3, 15, 27, 12, 3 divided by 60).
E = 0(0.05) + 1(0.25) + 2(0.45) + 3(0.20) + 4(0.05) = 0.25 + 0.90 + 0.60 + 0.20 = 1.95.
Answer: 1.95 televisions
Problem 14 Two Four-Sided Dice
There are 16 equally likely pairs. The sums 2, 3, 4, 5, 6, 7, 8 occur 1, 2, 3, 4, 3, 2, 1 times.
| x | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|
| P(x) | 116 | 216 | 316 | 416 | 316 | 216 | 116 |
E(X) = 2 + 6 + 12 + 20 + 18 + 14 + 816 = 8016 = 5. P(X ≥ 6) = 3 + 2 + 116 = 616 = 38.
Answer: E(X) = 5 and P(X ≥ 6) = 38
Problem 15 Food Truck Location
Park A: 0.6(900) + 0.4(300) = 540 + 120 = $660.
Plaza B: 0.6(600) + 0.4(550) = 360 + 220 = $580.
Park A has the higher expected earnings by $80. Plaza B has less variation, so it is the safer choice if the owner dislikes risk.
Answer: Park A ($660) beats Plaza B ($580) on expected value
Problem 16 Insurer’s Profit
Expected claim per policy = 0.02(3,000) = $60.
Expected profit per policy = 90 − 60 = $30.
Expected total profit = 5,000 × 30 = $150,000.
Answer: $60 per policy claim cost; $30 profit per policy; $150,000 in total
Problem 17 Guessing on a Quiz
(a) E = 4(15) + (−1)(45) = 45 − 45 = 0, so guessing is a fair gamble.
(b) E = 4(13) + (−1)(23) = 43 − 23 = 23 ≈ 0.67.
With three choices left the expected score is positive, so guessing helps on average. With five choices, guessing neither helps nor hurts.
Answer: (a) 0; (b) 23 ≈ 0.67, so guess in case (b)
Problem 18 Screening Parts
Without screening: 0.04(25) = $1.00 per part in expected refunds.
With screening: $0.50 per part and no refunds.
Screening saves 1.00 − 0.50 = $0.50 per part on average, so screening is recommended.
Answer: $1.00 versus $0.50 per part; screen the parts
Problem 19 Coin Flip Game
Heads 0, 1, 2, 3 have probabilities 18, 38, 38, 18, so the winnings $0, $2, $4, $6 have those probabilities.
E(winnings) = 0(18) + 2(38) + 4(38) + 6(18) = 248 = $3.
Expected net gain = 3 − 4 = −$1.
Answer: expected winnings $3; expected net gain −$1
Problem 20 Fundraiser Drawing
(a) Total prizes: 1,000 + 4(150) + 20(25) = 1,000 + 600 + 500 = $2,100. Expected winnings per ticket = 2,100400 = $5.25. Expected net gain = 5.25 − 10 = −$4.75.
(b) Money collected: 400(10) = $4,000. Profit = 4,000 − 2,100 = $1,900.
Answer: (a) $5.25 expected winnings, −$4.75 net; (b) $1,900 profit
Mis à jour le