Statistics Quiz: Analyzing Decisions And Strategies With Probability

What this quiz covers

This quiz focuses on Analyzing Decisions And Strategies With Probability, 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. Choose Option A, because $95\%$ is close to certainty and higher sensitivity always outweighs other concerns.
2. Choose Option A, because the harm of fraud is large, so probabilities are less important than avoiding a big loss.
3. Choose Option B, because if an order is flagged it must be fraudulent most of the time.
4. Choose Option B, because fraud is rare and the much lower false-flag rate will keep delays low while still meeting the $70\%$ fraud-caught requirement. (correct answer)

Explanation: This problem requires analyzing decisions using probability to balance fraud detection with customer experience. The retailer's goal is to keep the fraction of all orders delayed as low as possible while catching at least 70% of fraudulent orders. With 1% fraud rate, Option A flags (0.95 × 0.01) + (0.10 × 0.99) = 0.0095 + 0.099 = 10.85% of all orders. Option B flags (0.75 × 0.01) + (0.02 × 0.99) = 0.0075 + 0.0198 = 2.73% of all orders. Option B meets the 70% fraud detection requirement (75% > 70%) while flagging far fewer orders overall (2.73% vs 10.85%). Choice A incorrectly assumes higher sensitivity always outweighs other concerns, ignoring the impact on legitimate customers. Remember that probabilities guide decisions but don't guarantee specific outcomes. The strategy is to calculate total impact across all orders, not just focus on one metric.

Question 2

1. Strategy 2, because flagged attempts include many false alarms, so adding 2FA reduces risk without assuming every flag is truly malicious. (correct answer)
2. Strategy 3, because malicious logins are scary, so it’s best to block anything that looks suspicious regardless of the false-flag rate.
3. Strategy 1, because only $0.5\%$ of attempts are malicious overall, so flags are probably harmless.
4. Strategy 3, because $92\%$ detection means a flagged attempt is malicious $92\%$ of the time.

Explanation: This question involves the skill of analyzing decisions using probability. The company's goal is to minimize the chance that a malicious login attempt is allowed through, while avoiding treating every user as malicious. The relevant probabilities include a 0.5% base rate of malicious attempts, 92% detection of malicious ones, and 4% false flags on legitimate attempts, resulting in flagged attempts being malicious only about 10% of the time. Requiring 2FA for flagged attempts best aligns with this goal because it adds security to suspicious logins, stopping most malicious ones among them while inconveniencing but not blocking legitimate users. One incorrect option is to block all flagged attempts, which commits the flaw of base rate neglect by equating the 92% detection rate with the probability that a flag is malicious, leading to over-blocking good users. Remember that probabilities inform decisions by weighing risks, but they do not guarantee outcomes for any individual attempt. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities that matter most, such as the posterior probability of malice given a flag.

Question 3

1. Ignore the result, because $90\%$ specificity means positives are usually wrong in any population.
2. Start treatment immediately, because the condition is serious, so any chance is too high to wait for another test.
3. Start treatment immediately, because a positive result means the patient has the condition with $80\%$ probability.
4. Order a confirmatory test, because with a low base rate ($3\%$) and imperfect specificity, many positives can be false positives, so confirmation reduces unnecessary treatment. (correct answer)

Explanation: This question involves the skill of analyzing decisions using probability. The clinic's goal is to minimize unnecessary treatment while still taking positive results seriously. The relevant probabilities include a 3% base rate of the condition, 80% sensitivity, and 90% specificity, meaning a positive test indicates only about a 20% chance of the condition due to false positives. Ordering a confirmatory test best aligns with this goal because it verifies the initial positive, reducing the risk of treating false positives while addressing true cases. One incorrect option is to start treatment immediately, which commits the flaw of base rate neglect by assuming the 80% sensitivity means an 80% chance of the condition given a positive. Remember that probabilities inform decisions by quantifying diagnostic accuracy, but they do not guarantee outcomes for any patient. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities that matter most, such as the positive predictive value in low-prevalence settings.

Question 4

1. Position the spare at Route A, because the higher issue probability makes it more likely the spare will be needed and able to prevent a cancellation. (correct answer)
2. Position the spare at Route B, because once Route B had an issue on a day when no spare was available, causing a major disruption.
3. Position the spare at Route B, because Route B has fewer issues so the spare will be in better condition when needed.
4. Alternate days between A and B, because fairness is the best way to maximize prevention probability.

Explanation: This question involves the skill of analyzing decisions using probability. The airline's goal is to maximize the probability of preventing at least one cancellation using the spare aircraft. The relevant probabilities include a 6% issue rate for Route A and 3% for Route B, with the spare preventing a cancellation 90% of the time if positioned where an issue occurs. Positioning the spare at Route A best aligns with this goal because it yields a higher chance of successful prevention (5.4%) compared to Route B (2.7%), due to A's higher issue likelihood. One incorrect option is positioning at Route B based on an anecdote, which commits the flaw of anecdotal reasoning by prioritizing a single past event over probabilistic data. Remember that probabilities inform decisions by estimating expected benefits, but they do not guarantee outcomes on any given day. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities that matter most, such as the product of issue rates and prevention success.

Question 5

1. Policy 1, because offering to everyone guarantees that all would-be cancellers see it.
2. Policy 2, because $70\%$ of flagged users will cancel, since the model catches $70\%$ of cancellers.
3. Policy 3, because $15\%$ false flags means the model is unreliable, so targeting cannot work.
4. Policy 2, because the high-risk flag concentrates would-be cancellers more than the overall $10\%$ base rate, even though some flagged users would not cancel. (correct answer)

Explanation: This question involves the skill of analyzing decisions using probability. The company's goal is to target the retention offer so that most recipients are genuinely at risk of canceling, while still reaching many would-be cancellers. The relevant probabilities include a 10% base cancellation rate, 70% flagging of cancellers, and 15% false flags on non-cancellers, making flagged users about 34% likely to cancel versus 10% overall. Policy 2 best aligns with this goal because it concentrates offers on the higher-risk group (34% cancellation rate), reducing waste while covering 70% of potential cancellers. One incorrect option is Policy 1, which commits the flaw of not targeting by offering to all, resulting in only 10% of recipients being at risk and more wasted offers. Remember that probabilities inform decisions by assessing targeting efficiency, but they do not guarantee outcomes for any user. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities that matter most, such as the positive predictive value of the flag for cancellation risk.

Question 6

1. Require two recommendations, because it reduces false placements by making it less likely a non-benefiting student is recommended twice. (correct answer)
2. Use one quiz, because requiring two quizzes would unfairly exclude some students who could benefit.
3. Require two recommendations, because if a student is recommended twice, that guarantees they will benefit.
4. Use one quiz, because 80% accuracy among those who benefit means most placed students will benefit.

Explanation: This problem involves analyzing decisions using probability to decide on workshop placement criteria. The decision goal is to increase the likelihood that a placed student truly benefits. The key probabilities are the 15% base benefit rate and the quiz's 80% true positive and 20% false positive rates, with two quizzes being independent. Requiring two recommendations best aligns with the goal because it raises the positive predictive value to about 74% versus 41% for one quiz by reducing false positives. One incorrect option is D, which overstates the certainty, assuming two positives guarantee benefit, a flaw in misunderstanding conditional probability. Remember, probabilities inform better placement but do not guarantee every placed student benefits. To transfer this strategy, identify the goal, then compare predictive values for different thresholds.

Question 7

1. Strategy 1, because it flags fewer undamaged items, so it must also release fewer damaged items.
2. Strategy 1, because damage is rare (4%), so the difference between 70% and 90% detection will not matter in practice.
3. Strategy 2, because it catches a higher fraction of damaged items, reducing the chance damaged items slip through uninspected. (correct answer)
4. Strategy 2, because one time a thorough scan found damage that a quick scan missed, so it is safer.

Explanation: This problem involves analyzing decisions using probability to select the optimal screening strategy for damaged items. The decision goal is to minimize the probability that an item released without inspection is actually damaged. The key probabilities include a 4% base rate of damage, Strategy 1's 70% detection rate and 8% false positive rate, and Strategy 2's 90% detection rate and 18% false positive rate. Strategy 2 best aligns with the goal because its higher detection rate results in a lower conditional probability of damage given release (about 0.5% versus 1.34% for Strategy 1), prioritizing fewer missed damages over more inspections. One incorrect option is C, which commits the base rate fallacy by downplaying the detection difference due to rarity without calculating the actual impact on released items. Remember, these probabilities guide the decision to minimize risk but do not guarantee that no damaged items will ever be released. To transfer this strategy, identify the goal, then compare the conditional probabilities like P(damaged | released) that matter most.

Question 8

1. Place the crew downtown, because 60% is greater than 65%, so downtown is always the best bet.
2. Place the crew in the suburbs, because a friend once saw far more flats in the suburbs than downtown.
3. Place the crew downtown, because downtown has the higher share of reports on the more likely (weekday-like) day type. (correct answer)
4. Place the crew in the suburbs, because weekends have more reports in suburbs and the event could make it feel like a weekend.

Explanation: This problem involves analyzing decisions using probability to decide crew placement for flat-tire reports. The decision goal is to maximize the probability that the crew is in the area with the most reports tomorrow. The relevant probabilities are a 70% chance of weekday-like demand (60% downtown, 40% suburbs) and 30% chance of weekend-like (35% downtown, 65% suburbs). Placing the crew downtown best aligns with the goal because it matches the higher-probability weekday scenario where downtown has the most reports, yielding a 70% overall success probability versus 30% for suburbs. One incorrect option is C, which incorrectly compares percentages without weighting by scenario likelihood, a flaw in ignoring expected values. Remember, probabilities inform the best placement but do not guarantee the crew will always be in the right area. To transfer this strategy, identify the goal, then compare the weighted probabilities of success for each option.

Question 9

1. Plan B, because defective batches are only 10%, so missing some defects is not a big risk.
2. Plan A, because it catches a larger fraction of defective batches, so fewer defective batches will be released. (correct answer)
3. Plan A, because 80% means any batch not held has only a 20% chance of being defective.
4. Plan B, because it almost never holds good batches, so it is more reliable overall.

Explanation: This problem involves analyzing decisions using probability to choose a sampling plan for batches. The decision goal is to minimize the probability that a released batch is actually defective. The key probabilities are the 10% base defective rate, Plan A's 80% true positive and 5% false positive rates, and Plan B's 60% true positive and 1% false positive rates. Plan A best aligns with the goal because its higher detection rate results in a lower conditional probability of defect given release (about 2.3% versus 4.3% for B). One incorrect option is D, which downplays the detection difference due to low base rate, committing a base rate fallacy without calculating risks. Remember, probabilities guide minimizing release risks but do not guarantee no defective batches are released. To transfer this strategy, identify the goal, then compare conditional probabilities like P(defective | released) that matter most.

Question 10

1. Filter X, because 95% accuracy on malicious emails means a flagged email has a 95% chance of being malicious.
2. Either filter, because malicious emails are rare, so quarantine decisions are mostly irrelevant.
3. Filter X, because it catches more malicious emails, so quarantined emails will usually be truly malicious.
4. Filter Y, because with a 1% base rate, reducing false positives greatly increases the fraction of quarantined emails that are truly malicious. (correct answer)

Explanation: This problem involves analyzing decisions using probability to select an email filter. The decision goal is to maximize the probability that a quarantined email is actually malicious. The key probabilities are the 1% base rate of malicious emails, Filter X's 95% true positive and 4% false positive rates, and Filter Y's 85% true positive and 1% false positive rates. Filter Y best aligns with the goal because its lower false positive rate leads to higher precision (about 46% versus 19% for X), making quarantined emails more likely malicious despite catching fewer overall. One incorrect option is C, which confuses sensitivity with positive predictive value, a common misinterpretation of conditional probabilities. Remember, probabilities guide the decision to optimize review efficiency but do not guarantee all flagged emails are malicious. To transfer this strategy, identify the goal, then compare the positive predictive values that directly impact it.

Question 11

1. Either app is equally good because both have at least one situation where they do better than the other.
2. App S is best because $0.75$ is higher than $0.70$, so its overall performance must be higher too.
3. App H is best because it performs much better on the more common non-heavy-traffic days, which can outweigh App S’s advantage during heavy traffic. (correct answer)
4. App S is best because it has a higher on-time probability on heavy-traffic days, which are the only days that matter.

Explanation: This problem requires analyzing decisions using probability to maximize overall on-time delivery across all days. The company's goal is to maximize the overall probability of on-time delivery, not just on heavy-traffic days. For App H: (0.40 × 0.70) + (0.60 × 0.93) = 0.28 + 0.558 = 0.838 overall on-time probability. For App S: (0.40 × 0.75) + (0.60 × 0.88) = 0.30 + 0.528 = 0.828 overall on-time probability. App H's 83.8% overall performance exceeds App S's 82.8% because its much better performance on the more common non-heavy days (60% of time) outweighs App S's small advantage on heavy days. Choice C incorrectly compares single probabilities without weighting by frequency. Remember that probabilities inform decisions but don't guarantee specific outcomes. The strategy is to weight performance by how often each condition occurs to find overall effectiveness.

Question 12

1. Use Test X, because any false flag is acceptable if it prevents even one out-of-spec batch from shipping.
2. Use whichever test a supervisor prefers, since probabilities don’t determine what will happen on any particular batch.
3. Use Test X, because it catches more out-of-spec batches and therefore must be the best choice overall.
4. Use Test Y, because it meets the $70\%$ detection requirement and has a much lower false-flag rate, reducing unnecessary rework. (correct answer)

Explanation: This problem involves analyzing decisions using probability to minimize unnecessary rework in manufacturing. The factory's goal is to minimize unnecessary rework while ensuring at least 70% of out-of-spec batches get reworked. Test X detects 85% of out-of-spec batches but falsely flags 12% of in-spec batches, while Test Y detects 70% of out-of-spec batches but falsely flags only 4% of in-spec batches. Since Test Y meets the 70% detection requirement exactly and reduces false flags from 12% to 4% on the 85% of batches that are in-spec, it significantly reduces costly unnecessary rework. Choice A incorrectly prioritizes maximum detection without considering the cost of false positives. Probabilities inform decisions but don't determine what happens with any particular batch. The key is identifying the goal (minimize false flags while meeting detection threshold) and comparing the relevant probabilities.

Question 13

1. Use Method 1, because even one missed defective shipment would be unacceptable, regardless of how many false flags occur.
2. Use Method 1, because $90\%$ is higher than $70\%$, so it is always better at screening.
3. Choose whichever method the warehouse manager used successfully once before, since it worked in that case.
4. Use Method 2, because the defect is rare and Method 2 has a much lower false-flag rate while still catching more than half of defective shipments. (correct answer)

Explanation: This problem requires analyzing decisions using probability to choose a screening method. The company's goal is to minimize unnecessary manual inspections while still catching at least half of defective shipments. Method 1 catches 90% of defects but flags 8% of defect-free shipments, while Method 2 catches 70% of defects but flags only 2% of defect-free shipments. Since defects are rare (2%), Method 2 meets the requirement of catching at least 70% of defects while dramatically reducing false flags from 8% to 2% on the 98% of shipments that are defect-free. Choice A incorrectly focuses only on the detection rate without considering false flags. Remember that probabilities inform decisions but don't guarantee specific outcomes. The key strategy is to identify the goal (minimize false flags while meeting detection threshold) and compare the probabilities that matter most for that goal.

Question 14

1. Shoot left, because $0.78$ is the higher scoring probability on the initial shot and rebounds are too uncertain to matter.
2. Shoot left, because if a shot is saved then scoring is impossible, so only the initial scoring probability matters.
3. Shoot right, because the higher rebound scoring chance after a save can make the overall chance of scoring higher even if the initial scoring chance is slightly lower. (correct answer)
4. Shoot right, because the player scored once shooting right in a recent match, so that side is clearly best.

Explanation: This problem involves analyzing decisions using probability to maximize scoring chances in soccer. The coach's goal is to maximize the chance the team scores on the entire sequence (initial shot plus possible rebound). For shooting left: 0.78 initial + (1-0.78) × 0.20 = 0.78 + 0.044 = 0.824 total probability. For shooting right: 0.74 initial + (1-0.74) × 0.35 = 0.74 + 0.091 = 0.831 total probability. Since 0.831 > 0.824, shooting right gives a higher overall scoring probability despite the lower initial shot success rate. Choice A incorrectly ignores the rebound opportunity, focusing only on the initial shot probability. Probabilities inform optimal strategies but don't guarantee the outcome of any specific penalty kick. The transfer strategy is to identify the complete goal (total scoring probability) and calculate all relevant pathways to success.

Question 15

1. Action A, because the base rate of incorrect labels is only 0.5%, so a flagged package is still probably correct.
2. Action B, because a flagged result is much more likely when a label is actually incorrect than when it is correct, so checking every flagged package best minimizes the chance of shipping an incorrect one. (correct answer)
3. Action C, because requiring two flags guarantees the label is incorrect, eliminating the need for manual checks most of the time.
4. Action D, because random sampling avoids overreacting to scanner errors and provides a fair picture of overall quality.

Explanation: This question involves the skill of analyzing decisions using probability to determine the best action for flagged packages in a warehouse. The decision goal is to minimize the risk of shipping a mislabeled package, even if it requires extra manual checks. The relevant probabilities include a 0.5% base rate of incorrect labels, 92% sensitivity for flagging incorrect labels, and 2% false positive rate, resulting in a positive predictive value of about 19%, meaning flagged packages have a notable chance of being incorrect. Action B best aligns with these probabilities by manually verifying every flagged package, ensuring that the 19% truly incorrect ones are caught without letting them ship, while accepting the cost of checking false positives to minimize shipping errors. In contrast, Action A ships flagged packages anyway, relying on the low base rate overall but neglecting the elevated risk given a flag, a flaw known as base rate fallacy. Remember that probabilities inform the likelihood of a label error but do not guarantee whether any specific package is mislabeled. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities of true errors versus false alarms to prioritize risk reduction.

Question 16

1. Policy 1, because a positive screening result means the charger is almost certainly defective, so skipping teardown would be too risky.
2. Policy 3, because the overall defect rate is low, so the screening test result does not provide useful information for Standard chargers.
3. Policy 2, because with a low base defect rate in Standard chargers and a nonzero false positive rate, many flagged units will be fine; a confirmatory test sharply reduces unnecessary teardowns while still catching most true defects. (correct answer)
4. Policy 4, because Fast chargers have higher defect rates, so any flagged Standard charger can be assumed non-defective compared with Fast chargers.

Explanation: This question involves the skill of analyzing decisions using probability to choose the best policy for handling flagged Standard chargers. The decision goal is to minimize unnecessary destructive teardowns while still catching most truly defective units. The relevant probabilities show that for Standard chargers with a 1% defect rate, the screening test has a 90% sensitivity and 5% false positive rate, leading to a positive predictive value of about 15%, meaning most flagged units are not defective. Policy 2 best aligns with these probabilities by using a confirmatory test with 95% sensitivity and 1% false positive rate, which reduces unnecessary teardowns on false positives while catching nearly all true defects through the two-step process. In contrast, Policy 1 involves tearing down every flagged charger, which ignores the low positive predictive value and results in many unnecessary teardowns, a flaw known as base rate neglect. Remember that probabilities inform the likelihood of defects but do not guarantee any specific outcome for an individual charger. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities of true positives versus false positives across options to balance risks effectively.

Question 17

1. Discard immediately, because 97% positive on spoiled milk means a positive result is essentially proof of spoilage.
2. Send positives to the lab, because with a 1% base rate and an 8% false positive rate, many positive strip results are likely false; confirming reduces unnecessary discards while still catching true spoilage. (correct answer)
3. Ignore the strip entirely, because spoiled batches are rare and the shop can rely on smell tests instead.
4. Discard only if the strip is negative, because negatives happen 92% of the time for good milk and therefore are more informative.

Explanation: This question involves the skill of analyzing decisions using probability to manage positive test results for milk spoilage. The decision goal is to avoid discarding good milk unnecessarily while preventing spoiled milk from being served, with a 1% base spoilage rate, 97% sensitivity, and 8% false positive rate. These probabilities result in a positive predictive value of about 11%, meaning most positives are false. Sending positives to the lab best aligns with this by confirming results accurately, reducing unnecessary discards of false positives while catching true spoilage with high reliability. In contrast, discarding immediately (option A) treats the 97% sensitivity as proof of spoilage, ignoring the low positive predictive value due to the rare base rate, a flaw of base rate neglect. Remember that probabilities inform the chance of actual spoilage but do not guarantee the quality of any batch. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities of true positives versus false positives to decide on confirmation steps.

Question 18

1. Shoot right every time, because being unpredictable is more important than any percentage.
2. Shoot left every time, because 78% is higher than 74%.
3. Always repeat whatever worked in the last match, since recent success is the best guide.
4. Mix left and right roughly evenly, because it preserves the higher left success rate without triggering the keeper’s adjustment. (correct answer)

Explanation: This problem involves analyzing decisions using probability to choose a penalty-kick strategy. The decision goal is to maximize the chance of scoring on this kick. The relevant probabilities are the 78% success rate for left shots and 74% for right when mixed evenly, with a reduction to 70% for left if left is chosen more than half the time due to predictability. Mixing left and right roughly evenly best aligns with the goal because it avoids triggering the keeper's adjustment, allowing access to the higher 78% rate by choosing left without predictability. One incorrect option is A, which overlooks the predictability adjustment, a flaw in ignoring conditional probabilities based on strategy. Remember, probabilities inform the likely best strategy but do not guarantee a score on any given kick. To transfer this strategy, identify the goal, then compare the effective success probabilities under different behavioral adjustments.

Question 19

1. Choose the long ball because it is more exciting and sometimes produces dramatic goals, which can change momentum.
2. Choose the direct shot because it has the highest scoring probability, and the other options are therefore inferior regardless of counterattack risk.
3. Choose the short pass + cross because its scoring chance is only slightly lower than the long ball but it has a much lower counterattack-shot probability, aligning with the stated risk preference. (correct answer)
4. Choose the long ball because 7% is close to 8%, so the counterattack risk can be treated as essentially the same across strategies.

Explanation: This question involves the skill of analyzing decisions using probability to select the optimal free kick strategy in soccer. The decision goal is to maximize the chance of scoring while keeping the chance of conceding a shot on a counterattack as low as possible, preferring options with similar scoring chances but lower counterattack risk. The relevant probabilities are a 8% scoring chance for a direct shot (with unspecified but potentially high counter risk), 6% scoring with 1% counterattack for short pass + cross, and 7% scoring with 5% counterattack for long ball. The short pass + cross best aligns with these probabilities because its scoring chance is similar to the long ball's but with much lower counterattack risk, directly matching the risk preference. In contrast, choosing the long ball (option D) treats the 7% and 8% scoring as essentially equal while ignoring the higher counter risk, a flaw of overlooking the stated preference for lower risk in similar scenarios. Remember that probabilities inform the expected success of strategies but do not guarantee scoring or preventing counterattacks in any given play. To apply this strategy elsewhere, first identify the decision goal, then compare the probabilities of success and associated risks to find the best balance.

Question 20

1. Use Strategy A, because if an email is marked urgent then it is urgent $80\%$ of the time.
2. Use Strategy B, because urgent emails are only $30\%$ anyway, so it does not matter which strategy is used.
3. Use Strategy A, because it finds more urgent emails and capacity concerns should not affect the decision.
4. Use Strategy B, because it still captures at least $60\%$ of urgent emails and greatly reduces false urgents, helping prevent overload. (correct answer)

Explanation: This problem involves analyzing decisions using probability to manage email routing capacity. The team's goal is to keep the fast lane from being overloaded while capturing at least 60% of urgent emails. With 30% urgent emails, Strategy A sends (0.80 × 0.30) + (0.25 × 0.70) = 0.24 + 0.175 = 41.5% of all emails to fast lane. Strategy B sends (0.65 × 0.30) + (0.08 × 0.70) = 0.195 + 0.056 = 25.1% of all emails to fast lane. Strategy B meets the 60% urgent capture requirement (65% > 60%) while reducing fast lane volume from 41.5% to 25.1%, preventing overload. Choice A incorrectly ignores capacity constraints, focusing only on detection rate. Probabilities guide decisions but don't guarantee specific outcomes. The key is balancing detection goals with system capacity by calculating total impact across all emails.