Statistics Quiz: Compare Strategies Using Expected Value

What this quiz covers

This quiz focuses on Compare Strategies Using Expected Value, giving you a quick way to practice the rules, question types, and explanations that matter most for Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. Strategy B, because it treats the two outcomes as equally likely and averages $26 and$-4.
2. Strategy A, because it is more likely to give a positive net payoff (0.70).
3. Strategy A, because its expected net payoff is higher over many repetitions. (correct answer)
4. Strategy B, because it has the higher maximum net payoff ($26).

Explanation: This question involves comparing strategies using expected value in statistics and probability. Expected value represents the long-run average outcome if the strategy is repeated many times. For each strategy, compute the expected value by multiplying each possible net payoff—item value minus cost—by its probability and summing them up. In this case, Strategy A has a higher expected net payoff of $-0.50$ compared to Strategy B's $-1$. Over many trials, Strategy A would result in a higher average net payoff, favoring it for long-term gains. A tempting distractor is choosing Strategy B for its higher maximum net payoff of $26, but this ignores the low probability. To compare strategies, always multiply outcomes by their probabilities and sum to find and compare the totals, rather than looking at individual outcomes.

Question 2

1. Strategy A, because it has the greater expected value over many repetitions. (correct answer)
2. Strategy A, because it ignores the no-repair outcome and compares only $340$ vs $310$.
3. Strategy B, because the repair outcome still gives a positive payoff ($310$).
4. Strategy B, because paying more upfront must mean better coverage.

Explanation: This question involves comparing strategies using expected value in statistics and probability. Expected value represents the long-run average outcome if the strategy is repeated many times. For each strategy, compute the expected value by multiplying each possible net payoff—savings minus warranty cost—by its probability and summing them up. In this case, Strategy A has a higher expected net payoff of $0$ compared to Strategy B's $-30$. Over many trials, Strategy A would yield a higher average net payoff, making it preferable in the long run. A tempting distractor is assuming the pricier plan is better due to higher cost, but this overlooks the actual expected values. To compare strategies, always multiply outcomes by their probabilities and sum to find and compare the totals, rather than looking at individual outcomes.

Question 3

1. Strategy B, because it has the greater expected value over many attempts. (correct answer)
2. Strategy B, because earning points is more likely (0.80).
3. Strategy A, because it can earn as many as 40 points.
4. Strategy A, because its expected value is $0.30(40) + 0.70(10) = 19$ points.

Explanation: The skill here is comparing strategies using expected value, which helps determine the better choice for long-term average outcomes. Expected value is calculated as the long-run average points per attempt, found by multiplying each possible points outcome by its probability and summing them up. For Strategy A, the expected value is computed by weighting the outcomes of 40 and -10 by their probabilities of 0.30 and 0.70, resulting in 5. For Strategy B, weighting 8 and -5 by 0.80 and 0.20 gives 5.4. Since Strategy B has the higher expected value, it yields better average points over many attempts, though individual results may vary. A tempting distractor is favoring Strategy A for its maximum of 40 points, but expected value accounts for losses too. To apply this, always multiply outcomes by their probabilities and sum for each strategy, then compare the totals rather than isolated high or low values.

Question 4

1. Strategy A, because it has the greater expected value over many quizzes. (correct answer)
2. Strategy B, because gaining points is more likely (0.90).
3. Strategy B, because it has the larger maximum gain (4 points) compared to losing 6.
4. Strategy A, because its expected value is $0.50(12) + 0.50(6) = 9$ points.

Explanation: The skill here is comparing strategies using expected value, which helps determine the better choice for long-term average outcomes. Expected value is calculated as the long-run average score change per quiz, found by multiplying each possible change by its probability and summing them up. For Strategy A, the expected value is computed by weighting the changes of 12 and -6 by their probabilities of 0.50 and 0.50, resulting in 3. For Strategy B, weighting 4 and -20 by 0.90 and 0.10 gives 1.6. Since Strategy A has the higher expected value, it yields better average score change over many quizzes, though individual results may vary. A tempting distractor is choosing Strategy B for its high probability of gaining points, ignoring the severe loss impact. To apply this, always multiply outcomes by their probabilities and sum for each strategy, then compare the totals rather than isolated high or low values.

Question 5

1. Strategy A, because the $0 outcome is more likely in Strategy A.
2. Strategy A, because its expected value is $0.05(200) =$10$, which is greater than Strategy B.
3. Strategy A, because $200 is the largest possible profit.
4. Strategy B, because it has the greater expected profit over many emails. (correct answer)

Explanation: The skill here is comparing strategies using expected value, which helps determine the better choice for long-term average outcomes. Expected value is calculated as the long-run average profit per email, found by multiplying each possible profit by its probability and summing them up. For Strategy A, the expected value is computed by weighting the profits of $200 and$0 by their probabilities of 0.05 and 0.95, resulting in $10. For Strategy B, weighting$60 and $5 by 0.20 and 0.80 gives$16. Since Strategy B has the higher expected value, it yields better average profit over many emails, though individual results may vary. A tempting distractor is choosing Strategy A for its maximum profit of $200, overlooking the low probability. To apply this, always multiply outcomes by their probabilities and sum for each strategy, then compare the totals rather than isolated high or low values.

Question 6

1. Strategy B, because $10 is the largest single payoff.
2. Strategy A, because its expected value is $0.70(3)+0.20(8)=$3.7$, which is greater than Strategy B.
3. Strategy A, because winning $3 is the most likely outcome (0.70).
4. Strategy B, because it has the greater expected value over many spins. (correct answer)

Explanation: The skill here is comparing strategies using expected value, which helps determine the better choice for long-term average outcomes. Expected value is calculated as the long-run average winnings per spin, found by multiplying each possible winnings by its probability and summing them up. For Strategy A, the expected value is computed by weighting the winnings of $3,$8, and $0 by their probabilities of 0.70, 0.20, and 0.10, resulting in$3.70. For Strategy B, weighting $4,$10, and $1 by 0.50, 0.25, and 0.25 gives$4.75. Since Strategy B has the higher expected value, it yields higher average winnings over many spins, though individual results may vary. A tempting distractor is favoring Strategy A for its most likely outcome of $3, without summing all weighted values. To apply this, always multiply outcomes by their probabilities and sum for each strategy, then compare the totals rather than isolated high or low values.

Question 7

1. Strategy B, because its expected net value is higher over many repetitions. (correct answer)
2. Strategy A, because its expected net value is higher over many repetitions.
3. Strategy A, because its maximum prize ($50) is larger.
4. Strategy B, because it has a higher probability of winning a positive prize (0.45 vs 0.30).

Explanation: The skill here is comparing strategies using expected value to determine which raffle bundle is better in the long run. Expected value represents the long-run average net winnings if you buy the bundle many times. To compute the expected value for each strategy, multiply each possible net outcome (prize minus cost) by its probability and sum the results. In this case, Strategy B has a higher expected net value than Strategy A. Over many fundraisers, choosing Strategy B would lead to higher average net winnings, though results vary in any single purchase. A common distractor is selecting Strategy A due to its larger maximum prize of $50, but this ignores the low probability of winning it. To apply this elsewhere, always multiply outcomes by their probabilities and compare the totals rather than focusing on the best or worst single outcome.

Question 8

1. Strategy B, because its expected net savings per month are higher after subtracting the fee.
2. Strategy B, because the maximum possible savings ($50) is larger.
3. Strategy A, because its expected net savings per month are higher after subtracting the fee. (correct answer)
4. Strategy A, because it has a higher probability of saving money (0.40 vs 0.25).

Explanation: This problem asks us to compare membership strategies using expected value to find average net savings (savings minus fee) per month. Expected value tells us the long-run average outcome over many repetitions. For Strategy A (Basic): Expected savings = (0.40 × $20) + (0.60 ×$0) = $8, so net savings =$8 - $5 =$3 per month. For Strategy B (Plus): Expected savings = (0.25 × $50) + (0.75 ×$0) = $12.50, so net savings =$12.50 - $12 =$0.50 per month. Since Strategy A has higher expected net savings ($3 vs$0.50), it's the better choice over many months. A common error is focusing on the maximum possible savings ($50 in Strategy B) without considering its low probability and high membership fee. The key insight is to calculate expected gross savings first, then subtract the fixed monthly fee to find net expected value—the higher fee in Strategy B nearly eliminates its advantage from larger potential savings.

Question 9

1. Strategy A, because the most likely outcome is $120 (probability 0.70).
2. Strategy A, because its expected payment is higher than Strategy B’s.
3. Strategy B, because $300 is the largest possible payment.
4. Strategy B, because its expected payment is higher than Strategy A’s. (correct answer)

Explanation: This problem asks us to compare payment strategies using expected value, which tells us the average payment per job over many repetitions. Expected value is calculated by multiplying each outcome by its probability and summing the results. For Strategy A: EV = (0.70 × $120) + (0.30 ×$180) = $84 +$54 = $138 expected payment. For Strategy B: EV = (0.20 ×$300) + (0.80 × $110) =$60 + $88 =$148 expected payment. Since Strategy B has a higher expected value ($148 vs$138), it provides better average earnings over many jobs. A common mistake is choosing Strategy A because the most likely outcome ($120 with 70$300 more than compensates for its typically lower $110 payment.

Question 10

1. Strategy A, because its expected total cost is lower than Strategy B’s.
2. Strategy A, because paying $25 guarantees you never pay$300.
3. Strategy B, because its expected total cost is lower than Strategy A’s. (correct answer)
4. Strategy B, because most of the time (0.92) you pay $0.

Explanation: This problem asks us to compare strategies using expected value to find the average total cost over many years. Expected value tells us the long-run average outcome when facing uncertain events repeatedly. For Strategy A (Buy warranty): You always pay $25, and if failure occurs (probability 0.08), the warranty covers the$300, so total expected cost = $25. For Strategy B (No warranty): Expected cost = (0.08 ×$300) + (0.92 × $0) =$24 + $0 =$24. Since Strategy B has a lower expected total cost ($24 vs$25), it's the better choice over many years. A common mistake is thinking the warranty "guarantees" you never pay $300, making it seem safer, but this ignores that you're paying$25 every year regardless. The key insight is that insurance/warranties are profitable for sellers precisely because the expected cost without them is usually lower than the premium charged.

Question 11

1. Strategy A, because it guarantees you never pay $200.
2. Strategy B, because the most likely outcome is paying $0.
3. Strategy B, because its expected cost is $20, which is lower than$30. (correct answer)
4. Strategy A, because its expected cost is $10 (since damage is rare).

Explanation: This question involves the skill of comparing strategies using expected value to find which protection plan choice leads to lower average total cost over many buyers. Expected value is the long-run average outcome, representing typical total cost if situations repeat. For each, compute it by multiplying each possible total cost by its probability and summing. Strategy B has a lower expected cost of $20, compared to Strategy A's$30. Over many trials, Strategy B will result in lower average costs, despite risk of high repair bills. A tempting distractor is avoiding the worst-case $200, like in choice A, but expected value weighs the low damage probability. In practice, multiply all outcomes by probabilities to compare totals, not just guarantees or likeliest scenarios.

Question 12

1. Strategy A, because it has the larger maximum payoff ($20 item).
2. Strategy B, because it has a higher probability of a positive profit.
3. Strategy A, because its expected profit is $1.00 per play. (correct answer)
4. Strategy B, because its expected profit is $0.55 per play.

Explanation: This question involves the skill of comparing strategies using expected value to determine which loot-box option offers higher average profit over many plays. Expected value is the long-run average outcome, representing the typical profit per play if repeated many times. To compute it for each strategy, subtract the cost from each possible item value to get profits, then multiply each profit by its probability and sum the results. Strategy A has a higher expected profit of $1.00, while Strategy B has$0.85. Over many trials, Strategy A will provide a higher average profit, though individual plays may vary. A tempting distractor is focusing on the maximum payoff, like the $20 item in choice A, but this ignores probabilities of lower outcomes. To apply this skill, always multiply outcomes by their probabilities and compare the totals, rather than judging by single best or worst cases.

Question 13

1. Strategy A, because its expected net earnings are $6.00 versus$7.00 for Strategy B
2. Strategy B, because its expected net earnings are $7.00 versus$6.00 for Strategy A (correct answer)
3. Strategy A, because it has the higher probability of getting $10 in tips (0.80)
4. Strategy B, because it has the higher maximum tip amount ($12)

Explanation: This problem compares delivery route strategies using expected value, measuring net earnings (tips minus fuel cost). For Strategy A, we subtract the $2 fuel cost from expected tips: (0.80 ×$10) + (0.20 × $0) -$2 = $8 +$0 - $2 =$6.00. For Strategy B, with no extra fuel cost, the expected value is simply: (0.50 × $12) + (0.50 ×$2) = $6 +$1 = $7.00. Since$7.00 is greater than $6.00, Strategy B has the higher expected value over many deliveries. A tempting error is choosing Strategy A because it has a higher probability (0.80) of getting$10 in tips, but this ignores both the fuel cost and the probability weighting of all outcomes. The key is to calculate the complete expected net earnings by multiplying each outcome by its probability and subtracting all costs, not just focus on individual probabilities or maximum values.

Question 14

1. Strategy A, because its expected net profit is $0.20 versus$0.00 for Strategy B (correct answer)
2. Strategy B, because it has the larger maximum payout ($20)
3. Strategy A, because it has the higher probability of winning money (0.70 vs 0.20)
4. Strategy B, because its expected net profit is $4.00 versus$0.20 for Strategy A

Explanation: This problem asks us to compare strategies using expected value, which is the long-run average outcome when playing many times. For Strategy A, the expected value is calculated by multiplying each outcome by its probability: (0.70 × $6) + (0.30 ×$0) - $4 =$4.20 - $4 =$0.20 net profit. For Strategy B, the expected value is (0.20 × $20) + (0.80 ×$0) - $4 =$4 - $4 =$0 net profit. Since Strategy A has an expected value of $0.20 compared to$0 for Strategy B, Strategy A is favored over many trials. A common mistake is choosing Strategy B because it offers the larger maximum payout of $20, but this ignores that you only win 20% of the time. The key insight is to multiply outcomes by probabilities and compare totals, not just look at single outcomes.

Question 15

1. Strategy A, because its expected value is $18 versus$29 for Strategy B
2. Strategy B, because its expected value is $29 versus$18 for Strategy A (correct answer)
3. Strategy A, because it has the higher probability of a gain (0.60 vs 0.30)
4. Strategy B, because it has the larger maximum gain ($120)

Explanation: This problem asks us to compare trading strategies using expected value to find the higher average net profit. For Strategy A, the expected value is: (0.60 × $50) + (0.40 × -$30) = $30 + -$12 = $18. For Strategy B, the calculation is: (0.30 ×$120) + (0.70 × -$10) =$36 + -$7 =$29. Since $29 is greater than$18, Strategy B has the higher expected value over many trading days. A common mistake is choosing Strategy A because it has a higher probability of gain (0.60 vs 0.30), but this ignores the magnitude of gains and losses. Another error is focusing on Strategy B's larger maximum gain of $120 without considering probabilities. The transfer strategy is to multiply each outcome by its probability and sum them all, considering both positive and negative values to find the true long-run average.

Question 16

1. Strategy A, because the average total cost is $90 versus$45 for Strategy B
2. Strategy B, because the average total cost is $45 versus$90 for Strategy A (correct answer)
3. Strategy A, because it guarantees you never pay the $300 repair cost
4. Strategy B, because it has the higher probability (0.85) of costing $0

Explanation: This problem requires comparing strategies using expected value to find the lower average total cost. For Strategy A (buying warranty), you always pay $90 upfront, so the expected total cost is simply$90 regardless of whether the screen breaks. For Strategy B (no warranty), the expected cost is calculated as: (0.15 × $300) + (0.85 ×$0) = $45. Comparing these expected values, Strategy B has the lower average total cost at$45 versus $90 for Strategy A. Over many repetitions, choosing not to buy the warranty saves an average of$45 per phone. A tempting distractor is Strategy A because it guarantees you never pay the $300 repair cost, but this ignores that you're paying$90 every time regardless. The transfer strategy is to calculate the average cost by multiplying each outcome by its probability, not focus on avoiding the worst-case scenario.

Question 17

1. Strategy A, because the expected discount is $3.00 versus$0.30 for Strategy B
2. Strategy B, because the maximum discount is $30
3. Strategy B, because the expected discount is $3.00 versus$3.00 for Strategy A (correct answer)
4. Strategy A, because 90% of the time Strategy B costs $0 so it must have the higher expected value

Explanation: This problem asks us to compare coupon strategies using expected value, measuring the average discount cost to the store. For Strategy A, every customer gets $3 off, so the expected discount cost is$3.00. For Strategy B, we calculate the expected value: (0.10 × $30) + (0.90 ×$0) = $3 +$0 = $3.00. Both strategies have exactly the same expected value of$3.00 per customer, meaning they cost the store the same amount on average over many customers. A common mistake is choosing Strategy B because it offers the maximum discount of $30, ignoring that this only happens 10$0, but this reasoning incorrectly ignores the probability weighting. The key insight is that different probability distributions can have the same expected value when outcomes are properly weighted by their probabilities.

Question 18

1. Strategy A, because $1 off happens with certainty.
2. Strategy B, because it has the largest possible discount ($8).
3. Strategy B, because its expected discount is $1.60. (correct answer)
4. Strategy A, because its expected discount is $0.90.

Explanation: This question involves the skill of comparing strategies using expected value to determine which coupon approach costs the café more on average over many distributions. Expected value is the long-run average outcome, here the typical discount per coupon. Compute it for each by multiplying each discount amount by its probability and summing. Strategy B has a higher expected discount of $1.60, compared to Strategy A's$1.00. Over many trials, Strategy B will cost the café more on average, though individual discounts vary. A tempting distractor is preferring the certain low cost, like $1 in A as in choice A, but expected value includes potential higher discounts. To apply, calculate probability-weighted totals and compare, avoiding bias toward guarantees or maximums.

Question 19

1. Strategy A, because a donation is more likely (0.15 vs 0.10).
2. Strategy B, because it has the larger possible donation ($35).
3. Strategy A, because it has the greater expected net donation over many calls. (correct answer)
4. Strategy B, because its expected value is $0.10(35) - 2 =$5.5$.

Explanation: The skill here is comparing strategies using expected value, which helps determine the better choice for long-term average outcomes. Expected value is calculated as the long-run average net donation per call, found by multiplying each possible net donation by its probability and summing them up. For Strategy A, the expected value is computed by weighting the net donations of $20 -$1 and $0 -$1 by their probabilities of 0.15 and 0.85, resulting in $2. For Strategy B, weighting$35 - $2 and$0 - $2 by 0.10 and 0.90 gives$1.50. Since Strategy A has the higher expected value, it yields better average net donation over many calls, though individual results may vary. A tempting distractor is selecting Strategy B for its larger possible donation of $35, disregarding the costs and probabilities. To apply this, always multiply outcomes by their probabilities and sum for each strategy, then compare the totals rather than isolated high or low values.

Question 20

1. Strategy A, because 7 points is guaranteed so it must have the higher expected value
2. Strategy B, because its expected value is 10 points versus 7 points for Strategy A (correct answer)
3. Strategy B, because 20 points is the largest possible outcome
4. Strategy A, because Strategy B gives 0 points half the time so its expected value is 5 points

Explanation: This problem compares strategies using expected value to find the higher average points per day. For Strategy A, the expected value is simply 7 points since it's guaranteed every time. For Strategy B, we calculate the expected value by multiplying each outcome by its probability: (0.50 × 20) + (0.50 × 0) = 10 + 0 = 10 points. Since 10 points is greater than 7 points, Strategy B has the higher expected value over many repetitions. A tempting mistake is thinking Strategy A must be better because 7 points is guaranteed, but this ignores that Strategy B averages 10 points in the long run. Another error is thinking Strategy B's expected value is 5 points by incorrectly averaging 20 and 0, rather than properly weighting by probabilities. The transfer strategy is to multiply each outcome by its probability and sum them, not just compare single outcomes or guarantees.