After the light-path check, decide what the inventory numbers actually say
A carton has been checked, yet the classroom inventory system reports different results when a label is read twice. A manager asks whether a second check will reduce the chance of missing a problem and how many flagged cartons can be reviewed before the shift ends. The useful mathematical response is not ‘scan everything again’ without calculation. It is to build a probability model, test the assumptions behind it, and separate expected workload from guaranteed capacity.
This volume uses a fictional inventory project to practise simple events, combined events, tree diagrams, tables, sample counts, rates, percentages and contextual interpretation. The case follows the optical Physics in the preceding guide, but the mathematical owner is different. You will neither repeat ray-diagram calculations nor treat invented scan probabilities as a description of any real product.
Quick route: define one trial and its possible outcomes; state whether repetitions are assumed independent; multiply along the correct branches; add only mutually exclusive routes; interpret probabilities as model predictions rather than promises; compare demand with whole-number review capacity.
The examination is G3 Mathematics K310, not an integrated inventory paper
SEAB’s 2027 G3 syllabus listing identifies Mathematics K310, formerly 4052. The official K310 syllabus includes probability of single and simple combined events, including possibility diagrams and tree diagrams where appropriate, and work with statistical tables and graphs. Paper 1 and Paper 2 last 2 hours 15 minutes each. Each carries 90 marks and 50% of the examination weight. Paper 1 has about 26 short-answer questions; Paper 2 has 9–10 questions and ends with a real-world application problem. Essential working matters, and non-exact numerical answers are generally given to three significant figures unless instructed otherwise.
This chapter is an original question clinic for that syllabus, not an official sample paper. It builds on Vol 1285 Physics optical checking and Vol 1284 receiving-bay evidence. For G3 Mathematics study structure, see the Mathematics Paper 2 application guide. For younger learners building careful arithmetic and checking habits, use the Sengkang PSLE route.
One coherent case file, with every invented assumption visible
The fictional habitat has 120 labelled cartons. A preliminary single-check model assumes that 90% of independently repeated checks yield a readable result and 10% yield a flag. The numbers are teaching assumptions, not measured performance. A flagged carton needs one manual review taking five minutes. Two staff members are available for 30 minutes each, and both can review only one carton at a time. In a separate sample, researchers classify a set of cartons as correctly or incorrectly matched against a verified reference. Keep the probability model and the observed sample separate.
The choice of words matters. A ‘readable’ result is not necessarily a correct identification, and a ‘flag’ is not necessarily proof that a carton is wrong. ‘Independent’ is a mathematical assumption requiring justification in a real process. A rate calculated from old observations is not an automatic law of future results. Students who define these terms before doing arithmetic usually avoid mistakes hidden inside attractive percentages.
Use one row for each population
If a question says 120 cartons are in the inventory, 120 is the denominator when calculating a percentage of the whole inventory. If it says only 40 cartons were sampled, 40 is the denominator for a proportion in that sample. Dividing a sample count by the warehouse total without explanation creates a different statistic. Label a table with ‘all cartons’, ‘sampled cartons’, or ‘flagged cartons’ so the reference group cannot silently change.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Probability starts with a defined event
Let R mean a result is readable on a single check and F mean the result is flagged. Under the stated two-outcome model, P(R)=0.90 and P(F)=0.10. They add to one because this exercise defines them as exhaustive and mutually exclusive for a single trial. In real operations there may be more outcomes; a school model must say what has been simplified.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Branch multiplication needs a reason
If two checks are independent, P(R on both)=0.90×0.90=0.81. Independence means the result of one trial does not change the probability assigned to the next within the model. Do not multiply merely because there are two boxes on a diagram. A damaged label, a misaligned sensor or a persistent obstruction might make repeated checks dependent in practice. Good mathematics makes the assumption legible.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Union of routes is not the same as two independent events
The outcomes RR, RF, FR and FF cannot occur simultaneously for the same ordered pair. They are mutually exclusive routes through the same tree. The probability of exactly one flag is P(RF)+P(FR)=0.90×0.10+0.10×0.90=0.18. You add the probabilities of disjoint routes, not the individual probabilities of ‘readable’ and ‘flag’ without considering their overlap or the meaning of the event.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
An expected count is not a guaranteed count
If a model assigns 10% flags across 120 trials, 120×0.10=12 is an expected number. It does not mean that exactly twelve cartons must be flagged. The realised count may differ. A pupil who writes ’12 cartons will definitely need review’ has changed a model’s expectation into a certainty. When planning capacity, it is sensible to compare several possible observed counts.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Work in whole cartons and whole reviews
Two reviewers each have 30 minutes, and each review takes five minutes under the simplified model. One reviewer can finish six full reviews; together they can finish twelve. Even if a calculation produces 12.8, the team cannot complete a fractional carton review. Whole-number capacity is a physical count, so the final decision may need rounding down, with the reason stated.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Use tables for records, not assumptions
If a question provides verified reference labels and scanner outputs, make a table before interpreting accuracy. Correctly matched, wrongly matched, flagged and readable are different properties. The same result can be readable but incorrect. A one-way table can conceal this distinction; a two-way table makes it visible. Copy totals before calculating rates to detect entries that have been counted twice.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Sampling does not automatically remove bias
A student chooses ten cartons from the most accessible shelf and finds no label problems. That does not establish that the entire warehouse has no problems, because the selection might exclude difficult shelves or damaged labels. The G3 data-handling skills include clear interpretation and awareness that a diagram or small set of data can mislead. State the sample method when a broad claim is proposed.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
The inverse question is often harder
Instead of asking how many cartons can be reviewed, ask how many minutes are needed to review eighteen. At five minutes each, eighteen reviews require 90 person-minutes. Two people working in parallel need at least 45 clock minutes if work divides cleanly and there are no interruptions. This is different from saying one person needs 90 minutes, and it depends on both having work available throughout.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Describe a diagram in words as well as symbols
A tree with first-check branches R and F and second-check branches R and F must attach probabilities to the correct level. Write ‘readable first, flagged second’ beside RF. A reader then sees why the route has probability 0.09 instead of guessing from unlabelled arrows. The diagram is an argument about the event space, not an ornament to place beside a calculation.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
How a percentage point differs from a percent change
If the flag proportion rises from 10% to 15%, it rises by five percentage points. Relative to the original 10%, the increase is 50%. Both computations are correct, but they answer different questions. A report that says ‘increased by five percent’ without the intended comparison can mislead. In examination answers, name the baseline and the unit of change.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Bound a conclusion before recommending action
Suppose the study calculates a review capacity of twelve and the current observed queue has fourteen. The model shows a two-review shortfall under the stated time and staffing assumptions. It does not show the queue is unsafe, that a second shift is required by law or that any real device is defective. A proportionate fictional recommendation is to revise the queue plan and obtain a verified count.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Make the last Paper 2 question a sequence
Read the complete scenario once for the purpose, then list given quantities and their units. Solve subparts in order but check whether an earlier answer was only an estimate. When a later part asks about feasibility, compare whole-number resources and actual demand. In the final sentence, use the language of the scenario, not just ‘yes’ or ‘no’. The written judgement should depend on visible working.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Find the weakest link in an incorrect answer
If a learner draws a correct tree yet adds branch probabilities incorrectly, target addition of mutually exclusive routes. If the learner multiplies non-independent trials, target the modelling decision. If the probability is correct but the report claims certainty, target interpretation. Correction becomes quicker when the error is named precisely rather than counted only as one mark lost.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Accuracy is not the same as high confidence
A written answer can have many decimal places and still use the wrong denominator. Three-significant-figure rounding is a presentation rule, not a cure for conceptual error. Estimate first: P(RR) must be below 0.90 because two readable outcomes are more demanding than one. If a computed probability of both readable is 1.8, it is impossible. Basic bounds catch mistakes before a calculator has the final word.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Teach the transition rather than skipping it
Students moving from PSLE to secondary mathematics bring useful ratio, fraction and percentage skills. G3 probability extends that base into carefully defined outcomes and combinations. First write fractions in the familiar way, then introduce R, F and tree branches; finally move to a decision about review capacity. Advanced work grows from clear foundations, not from the rapid introduction of unfamiliar symbols.
For the inventory model, the immediate practice is to rewrite the conclusion with its denominator, event and assumption visible. Then test what changes if one quantity in the story changes. This small transfer step prevents a memorised numerical pattern from masquerading as an understanding of the problem.
Forty worked checks, from single events to a feasible shift decision
1. Complement
If P(R)=0.90, P(F)=0.10 because the two defined outcomes cover all possibilities.
Working check: Write the complete sample space before subtracting. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
2. One flag in ten
A model with probability 0.10 does not guarantee one flag in every sequence of ten checks.
Working check: Separate probability from realised frequency. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
3. Both readable
Independent checks yield P(RR)=0.90²=0.81.
Working check: A joint event is less likely than a single success. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
4. Both flagged
P(FF)=0.10²=0.01 under independence.
Working check: Do not accidentally write 0.20 by adding. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
5. Readable then flagged
P(RF)=0.90×0.10=0.09.
Working check: Order matters in describing the route. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
6. Flagged then readable
P(FR)=0.10×0.90=0.09.
Working check: Equal values do not make the routes identical. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
7. Exactly one flag
P(RF)+P(FR)=0.18.
Working check: The two routes cannot occur together, so add. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
8. At least one flag
1−P(RR)=1−0.81=0.19.
Working check: The complement often saves unnecessary enumeration. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
9. At least one readable
1−P(FF)=0.99.
Working check: The event includes RR, RF and FR. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
10. Verify the tree
0.81+0.09+0.09+0.01=1.00.
Working check: Tree leaf probabilities must total one. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
11. Convert to percent
A probability of 0.19 corresponds to 19%.
Working check: Convert only at the end if the question uses probabilities. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
12. From percentage to count
10% of 120 is 12 as a model expectation.
Working check: Do not call twelve an observed number. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
13. Observed queue
If 14 of 120 were actually flagged, the observed proportion is 14/120 ≈ 11.7%.
Working check: Use the observed count when reporting observed data. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
14. Comparing expectation
Expected 12 versus observed 14 is a difference of two cartons.
Working check: The comparison does not establish why the count differed. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
15. One reviewer’s capacity
30 minutes divided by five minutes each gives six complete reviews.
Working check: Include the unit reviews rather than only the number six. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
16. Two reviewers
Two independent reviewers working in parallel can do twelve within the 30-minute model.
Working check: If the task requires a single supervisor, the assumption changes. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
17. Shortfall
A queue of 14 and capacity of 12 leaves two unreviewed in this shift.
Working check: State the decision as conditional on the timing assumptions. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
18. Extra duration
Two remaining reviews need ten person-minutes or five clock minutes if two reviewers can take one each.
Working check: Person-minutes and elapsed minutes are not interchangeable. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
19. Review eighteen
Eighteen at five minutes each require 90 person-minutes.
Working check: Two workers need at least 45 minutes in the ideal parallel model. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
20. Three reviewers
Three staff at six reviews each have capacity eighteen in the same 30-minute period.
Working check: Describe whole-number throughput instead of a raw fraction. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
21. If each review is six minutes
One worker can do five reviews in 30 minutes. Two can do ten.
Working check: Change the unit rate before deciding capacity. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
22. Ninety percent of 50
An expected 45 readable single-check results and five flags follow the model.
Working check: Do not equate ‘readable’ with correctly labelled. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
23. Ten percent of 40
A four-flag expectation over forty hypothetical checks is not a statement about a particular sample.
Working check: Probabilities forecast patterns, not exact outcomes. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
24. A simple tally
Of 50 verified cases, 36 clean cartons are correctly read, four clean cartons flagged, seven mismatched cartons flagged and three mismatched cartons read without flag. Total 50.
Working check: Build the 2×2 count table before computing any rate. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
25. Total flagged
In that observed sample, four plus seven equals eleven flags.
Working check: The denominator for ‘fraction of flags that are actual mismatches’ is eleven. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
26. Fraction of flags that were mismatches
Seven of eleven flagged cases were verified mismatches, 7/11 ≈ 63.6%.
Working check: This is a description of the sample, not a verified device property. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
27. Fraction of sample mismatches
Ten of fifty sample cases were mismatches, 20%.
Working check: Never divide by the number of flags for a whole-sample fraction. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
28. Overall matching decision rate
The sample has 36 correctly cleared cases and seven correctly flagged mismatches, so 43/50=86% matched the reference.
Working check: Define the chosen success measure before calling it ‘accuracy’. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
29. Different denominators
7/10 is the fraction of mismatches flagged; 36/40 is the fraction of clean cases correctly cleared.
Working check: A ratio is meaningless without its named reference group. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
30. Tree event with 0.8
If P(R)=0.8 in a new hypothetical model, P(at least one readable in two independent trials)=1−0.2²=0.96.
Working check: Do not reuse 0.90 from an earlier model. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
31. Exactly one readable at 0.8
P(RF)+P(FR)=0.8×0.2+0.2×0.8=0.32.
Working check: Write separate ordered outcomes. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
32. One out of twenty
A simple random draw from 20 cartons with five marked needs review has probability 5/20=1/4.
Working check: This assumes equal chances of choosing each carton. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
33. Two draws without replacement
With five marked among twenty, probability of drawing two marked in a row without replacement is 5/20×4/19=1/19 ≈ 5.26%.
Working check: The second denominator changes because the carton was not replaced. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
34. Two draws with replacement
If the carton is returned and the sample is mixed, P(two marked)=5/20×5/20=1/16=6.25%.
Working check: Replacement makes the simplified probabilities constant. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
35. Percentage points
A flag rate from 10% to 15% rises by five percentage points.
Working check: Relative increase is 50%; label which you mean. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
36. Graph audit
A bar chart starting at 9% can exaggerate a change from 10% to 11%. Inspect the axis before describing its effect.
Working check: Graphs communicate through scale as well as numbers. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
37. Subgroup comparison
A shelf has 3 flags among 20, another 6 among 80. Their observed rates are 15% and 7.5%, not simply three versus six.
Working check: Compare proportions when subgroup sizes differ. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
38. Independent assumption questioned
If one dirty label causes both scans to fail, the two results may be dependent. The 0.81 independent formula may not describe them.
Working check: Do not treat repeated trials as automatically fresh independent draws. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
39. A capacity gate
Fifteen actual flags and twelve possible reviews mean the shift cannot clear the queue under the stated model.
Working check: A true inequality is not a safety certification. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
40. An evidence-safe recommendation
Report ‘Fourteen cartons are flagged in the current record; twelve can be reviewed in the scheduled period under the given rates.’
Working check: An exact, conditional statement is stronger than ‘almost all are fine’. Write the event and its reference group before applying an operation. If the result describes a hypothetical probability, do not quietly turn it into an actual count. If it describes recorded data, do not treat it as a universal law. The final sentence should say exactly which population, time period or model the calculation covers.
A complete Paper 2-style extended example
In an original practice problem, 120 cartons are checked. The model says each check independently has a 0.10 probability of being flagged. Exactly fourteen flags appear in the observed record. One manual review takes five minutes, and two staff can each devote 30 minutes to manual reviews. Part A: calculate the model probability that two independent checks are both readable. Part B: calculate the chance that exactly one of the two checks is flagged. Part C: calculate the observed flag percentage. Part D: determine whether this observed queue can be cleared in the given time. Part E: write a two-sentence evidence-based report.
Answer A: 0.90×0.90=0.81. Answer B: 0.90×0.10+0.10×0.90=0.18. Answer C: 14/120×100%=11.666…%, approximately 11.7%. Answer D: 30/5=6 reviews per person, so two people complete at most twelve; fourteen is greater than twelve and two cases remain. Answer E: ‘Fourteen cartons were flagged in the observed inventory record, equivalent to about 11.7% of the 120 cartons. Under the stated five-minute review rate and staffing time, only twelve cases can be reviewed in the current period, leaving two for a revised plan.’
A common weak answer would say that because the expected number of flags is twelve, the two reviewers can definitely finish. That confuses an expected count with the observed queue of fourteen. Another weak answer would call 0.81 an observed success rate when it is merely a probability computed from the independent two-check assumption. To gain reliable examination marks, name both distinctions. A careful mathematical reader is always asking: which number describes the model, and which one describes the record?
How to revise after marking
After each timed attempt, sort errors into four groups: event definition, probability structure, arithmetic and interpretation. For an event-definition error, redraw the tree and list the outcomes. For a probability-structure error, explain independence and mutually exclusive routes using words, not only letters. For arithmetic errors, use fractions first and estimate the expected size before calculating. For interpretation errors, rewrite the conclusion with the actual denominator and assumption. Spend most correction time on the first incorrect move. Repeat a new version the next day, changing at least one probability, count or time value.
At the advanced level, compare plans rather than numbers alone. One extra reviewer may increase ideal capacity, but only if the additional person has the required time and can review independently. A lower flag probability may reduce expected workload, but the actual recorded queue still determines how many manual checks exist. The ability to keep these claims separate transfers well to the final contextual question of Paper 2 and to more mature decision-making outside school.
Relevant learning routes
Review the preceding Physics light-path case for what the optical law can establish; the earlier integrated receiving-bay review for checking limits of evidence; the G3 Mathematics application owner for full Paper 2 habits; the Sengkang PSLE preparation route for arithmetic foundations; and the official K310 syllabus for the exact mathematics assessment requirements. A later English chapter will ask how a reader should interpret the way such records are presented in a notice.
