When a question says ‘at least one’, begin by reading what it includes
A bag contains three red counters and two blue counters. You select two counters, one after the other, without putting the first back. What is the probability that at least one counter is blue? You could draw a large tree, calculate blue–blue, blue–red and red–blue, then add the three probabilities. Or you could ask a simpler question: what is the probability of getting no blue counters? No blue means two red counters. That single ordered route has probability 3/5 × 2/4 = 3/10. Therefore the requested probability is 1 − 3/10 = 7/10. The idea is not a trick. It is a precise account of every possible outcome, arranged so that you count each one once.
Students often get a probability question wrong even after calculating every fraction correctly. They answer ‘exactly one’ when the question says ‘at least one’, treat a second draw as though an object was replaced, or count an ordered path twice. This chapter teaches how to read the event first, choose its complement and confirm that the chosen method matches the actual experiment. The method is a step beyond Volume 0258 on shifting denominators without replacement, and it builds on the earlier owners Volume 0178 on probability-tree routes and Volume 0182 on mutually exclusive versus independent events. We do not repeat those chapters’ broad curriculum; we take ownership of the narrower wording ‘at least one’ and its opposite event.
The published examination context
For the 2027 Singapore-Cambridge Secondary Education Certificate, G3 Mathematics is syllabus K310, following the previous O-Level Mathematics code 4052. Its Statistics and Probability strand covers probabilities of single events and simple combined events, including possibility diagrams and tree diagrams where appropriate, as well as rules for mutually exclusive and independent events. The complement of an event is a clear way to reason about simple combined outcomes. We do not treat advanced conditional-probability formulas, distributions or general combinatorics as compulsory K310 topics. Use the official 2027 SEC G3 syllabus index and SEAB K310 PDF for the exact examination requirements.
G3 Mathematics has two papers. Each lasts 2 hours 15 minutes, contributes 50% and carries 90 marks. Paper 1 has approximately 26 compulsory short-answer questions; Paper 2 has nine to ten questions and ends with a real-world mathematics application question. Relevant working should be shown because a correct number without necessary reasoning can lose credit. The original questions in this guide are practice examples, not published SEC questions or predictions. They are intentionally simple enough to keep attention on event logic, which is the transferable examination skill.
Three phrases that change the whole calculation
At least one means one or more: one, two or more occurrences as permitted by the number of trials. Exactly one means precisely one occurrence, excluding zero and excluding two or more. At most one means zero or one. These are different sets of outcomes. If two coins are tossed, the possibilities are HH, HT, TH and TT. ‘At least one head’ contains HH, HT and TH: three of four. ‘Exactly one head’ contains HT and TH: two of four. ‘At most one head’ contains HT, TH and TT: three of four. One missed word changes the event even though the same sample space is used.
Whenever ‘at least one A’ feels complicated, ask what would make that statement false. It is false only when no A occurs. The complement of ‘at least one blue’ is ‘no blue’. The complement of ‘at least one six’ is ‘no six’. The complement of ‘at least one success’ is ‘all attempts unsuccessful’. If there are only two draws, these opposites are usually easier to calculate because the ‘none’ event follows one repeated category across the trials. The complement gives P(A) = 1 − P(not A), but you must be precise about the definition of A.
Why subtraction from one works
All possible outcomes together have probability one. An event and its complement cannot occur together and between them cover every possible outcome. So their probabilities add to one. This is why one minus the probability of no blue produces the probability of at least one blue. The subtraction does not assume independence, replacement or a particular number of draws. Those details matter in computing the probability of the complement. It is important not to confuse the logical fact that an event and its complement cover the sample space with a numerical shortcut for the probabilities within a multi-stage route.
If you use a probability tree, multiplication follows a path and addition combines distinct mutually exclusive complete paths. The complement method often replaces the addition of several paths with subtraction of one more convenient event. Yet it still requires accurate branch counts. For the opening bag, drawing red reduces the red count from three to two and the total from five to four, giving P(no blue) = 3/5 × 2/4. If the red counter were replaced, that probability would instead be 3/5 × 3/5. The complement principle stays valid in both cases, but the numerical model changes.
A small checking table in words
For the three-red, two-blue bag without replacement, the four colour routes have probabilities: red then red, 3/5 × 2/4 = 3/10; red then blue, 3/5 × 2/4 = 3/10; blue then red, 2/5 × 3/4 = 3/10; blue then blue, 2/5 × 1/4 = 1/10. They total one. The at-least-one-blue event includes the last three routes and totals 3/10 + 3/10 + 1/10 = 7/10. The complement approach obtains the same answer by subtracting red-red from one. This independent check shows both completeness and the absence of double counting.
A useful boundary check is that a probability must lie between zero and one. For an event like ‘at least one success’, adding a new possible trial cannot make the event less likely if the process is otherwise fixed. A value above one or a change in the wrong direction usually signals incorrect addition, an incorrect event or a denominator that was not updated. Another check is to compare the complement with a single-trial probability. In a two-draw experiment with several eligible objects, ‘at least one’ is often more likely than success on just the first draw; if your calculation says otherwise, recheck the steps. These are reasonableness checks, not a substitute for correct working.
Worked questions: first understand the wording
Event language before arithmetic
Before reaching for a probability formula, translate the phrase into a set of possible outcomes. A single word such as ‘exactly’ or ‘most’ can reverse the event. A complement is useful when the opposite event is simpler, not because every problem must be solved that way.
Original probability clinic 01: A fair coin is tossed twice. Find the probability of at least one head.
Find the event. A fair coin is tossed twice. Find the probability of at least one head. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Four equal outcomes are HH, HT, TH, TT. Only TT has no head. Thus 1 − 1/4 = 3/4. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not omit HH: at least one includes two heads. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Write the event set explicitly and show how it differs from exactly one head. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 02: A fair coin is tossed twice. Find exactly one head.
Find the event. A fair coin is tossed twice. Find exactly one head. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. HT and TH are the two favourable ordered outcomes out of four, so the probability is 1/2. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. HH does not belong to exactly one even though it satisfies at least one. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Compare your answer with P(at least one head) = 3/4. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 03: A fair coin is tossed twice. Find at most one head.
Find the event. A fair coin is tossed twice. Find at most one head. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. The only excluded outcome is HH. Hence 1 − 1/4 = 3/4. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. At most one includes zero heads, so TT is favourable. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain why the numerical result matches at least one head while the events differ. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 04: A fair die is rolled once. Find the probability of at least one six.
Find the event. A fair die is rolled once. Find the probability of at least one six. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. There is only one trial; the event is simply rolling six, probability 1/6. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The phrase at least one does not create extra trials by itself. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. State the opposite event and use 1 − 5/6 as a cross-check. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 05: A fair die is rolled twice. Find the probability of no six.
Find the event. A fair die is rolled twice. Find the probability of no six. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Each roll has five non-six faces out of six, so P(no six on both) = (5/6)^2 = 25/36. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not subtract from one when the question already asks for the complement event. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain why the roll probabilities are unchanged between trials. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 06: A fair die is rolled twice. Find the probability of at least one six.
Find the event. A fair die is rolled twice. Find the probability of at least one six. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Complement the event of no six: 1 − (5/6)^2 = 11/36. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Adding 1/6 + 1/6 double counts outcomes with two sixes. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Show the three disjoint route types: six then non-six, non-six then six, six then six. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 07: A single card is drawn from an exclusive red, blue and yellow deck.
Find the event. A single card is drawn from an exclusive red, blue and yellow deck. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Red or blue is a union of exclusive single-card categories, so P(red or blue) = P(red) + P(blue). In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not use the complement of ‘neither red nor blue’ without first knowing the yellow proportion; both methods are equivalent. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain the difference between adding exclusive categories and multiplying sequential draw probabilities. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Without replacement: update the bag before subtracting
Every draw changes the composition when the selected object is kept outside the bag. Track both the desired category and the total population. Subtract from one only after the complement probability has been calculated from the correct series of remaining counts.
Original probability clinic 08: A bag has three red and two blue counters. Draw twice without replacement; at least one blue.
Find the event. A bag has three red and two blue counters. Draw twice without replacement; at least one blue. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No blue means red then red, 3/5 × 2/4 = 3/10. Answer: 7/10. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Using 3/5 twice wrongly assumes the first red is returned. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Verify by adding blue-red, red-blue and blue-blue routes. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 09: The same bag asks for exactly one blue in two draws.
Find the event. The same bag asks for exactly one blue in two draws. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. The two routes are BR and RB, each with probability 3/10, so exactly one blue is 3/5. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. At least one would also include BB, giving 7/10. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Write each ordered path to avoid treating the two possibilities as one. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 10: The same bag asks for both counters blue.
Find the event. The same bag asks for both counters blue. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Blue then blue has probability 2/5 × 1/4 = 1/10. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The complement of ‘no blue’ is not ‘both blue’. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Check that exactly one blue plus both blue equals the at-least-one result. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 11: A bag has four green and six yellow tokens. Draw two without replacement; at least one green.
Find the event. A bag has four green and six yellow tokens. Draw two without replacement; at least one green. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. P(no green) = 6/10 × 5/9 = 1/3; P(at least one green) = 2/3. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The yellow numerator falls because a yellow token was removed. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Name the sample space at the second draw and simplify the fraction. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 12: The same ten-token bag asks for exactly one green.
Find the event. The same ten-token bag asks for exactly one green. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Paths GY and YG are 4/10 × 6/9 and 6/10 × 4/9, totalling 48/90 = 8/15. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Omitting either ordering gives half the valid probability. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Compare with P(two green) = 4/10 × 3/9 = 2/15, so at least one is 10/15. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 13: A bag holds two blue and three black objects; three are drawn without replacement. At least one blue.
Find the event. A bag holds two blue and three black objects; three are drawn without replacement. At least one blue. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No blue means choosing all three black: 3/5 × 2/4 × 1/3 = 1/10. Therefore at least one blue is 9/10. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. A two-draw complement would be incomplete because the experiment has three draws. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Check the number of draws against the words in the question. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 14: A bag has four red and three blue objects. Three are drawn without replacement; at least one blue.
Find the event. A bag has four red and three blue objects. Three are drawn without replacement; at least one blue. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No blue means three reds: 4/7 × 3/6 × 2/5 = 4/35, so at least one blue is 31/35. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The product is not three copies of 4/7; counts shrink. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Use a simple tree to explain the repeated red path. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 15: A bag contains two gold and eight silver tickets. Two are drawn without replacement; at least one gold.
Find the event. A bag contains two gold and eight silver tickets. Two are drawn without replacement; at least one gold. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No gold means two silver: 8/10 × 7/9 = 28/45. The answer is 17/45. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not count favourable outcomes by subtracting the number of gold tickets from the total without defining pairs. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Check the answer by calculating gold-gold, gold-silver and silver-gold. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 16: A bag contains five black and five white counters. Draw three without replacement; at least one black.
Find the event. A bag contains five black and five white counters. Draw three without replacement; at least one black. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No black means three white: 5/10 × 4/9 × 3/8 = 1/12, so at least one black is 11/12. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Ignoring the third draw undercounts ways to see a black counter. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain why repeated sampling without replacement changes the chances of later colours. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Independent or with-replacement experiments
An independent trial has the same success or failure probability regardless of earlier outcomes in the stated model. Returning a drawn object to a thoroughly mixed bag restores its original composition. Use the given assumption rather than deciding independence from a convenient calculation.
Original probability clinic 17: A fair coin is tossed three times. At least one head.
Find the event. A fair coin is tossed three times. At least one head. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No head means TTT, probability (1/2)^3 = 1/8. The complement is 7/8. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Exactly one head would be only three of the eight routes, not seven. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Write down the one excluded route and explain why its probability is 1/8. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 18: A fair coin is tossed four times. At least one tail.
Find the event. A fair coin is tossed four times. At least one tail. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No tail means HHHH, probability (1/2)^4 = 1/16; answer 15/16. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The four trials do not mean four times 1/2, which could exceed one. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain why the no-tail event requires all four heads. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 19: A fair die is rolled three times. At least one six.
Find the event. A fair die is rolled three times. At least one six. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. P(no six) = (5/6)^3 = 125/216. Answer 91/216. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not add 1/6 three times; overlaps must be considered. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Estimate whether your answer is below or above 1/2 and check plausibility. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 20: Two independent attempts have a success chance 0.3 each. At least one succeeds.
Find the event. Two independent attempts have a success chance 0.3 each. At least one succeeds. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. P(both fail) = 0.7 × 0.7 = 0.49. P(at least one) = 0.51. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. A success chance of 0.3 for one attempt does not imply 0.6 for two attempts. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain the difference between two independent attempts and two exclusive events. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 21: Four independent attempts each have success probability 0.2. At least one succeeds.
Find the event. Four independent attempts each have success probability 0.2. At least one succeeds. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. P(all four fail) = 0.8^4 = 0.4096, so the answer is 0.5904. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The failure probability must be repeated four times because the event spans four trials. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Check that the result is greater than the single-attempt success chance 0.2. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 22: Three red and two blue counters are drawn twice with replacement. At least one blue.
Find the event. Three red and two blue counters are drawn twice with replacement. At least one blue. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No blue is two red draws, probability (3/5)^2 = 9/25, giving P(at least one blue) = 16/25. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The without-replacement answer of 7/10 belongs to a different experiment. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Identify the exact word that determines the second denominator. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 23: Two independent attempts each succeed with probability 0.4. Compare both, exactly one and at least one.
Find the event. Two independent attempts each succeed with probability 0.4. Compare both, exactly one and at least one. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Both succeed: 0.4² = 0.16. Exactly one: 2 × 0.4 × 0.6 = 0.48. At least one: 1 − 0.6² = 0.64. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The three values have different event meanings; exactly one excludes both successes. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Verify that 0.48 + 0.16 equals 0.64. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 24: Three independent checks have a 0.1 chance of finding an issue each. At least one finds one.
Find the event. Three independent checks have a 0.1 chance of finding an issue each. At least one finds one. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Assuming independent trials with unchanged chance, P(at least one) = 1 − 0.9³ = 0.271. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The numerical model depends on an explicit independence assumption, which may not hold for repeated checks of the same item. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. State the model assumption without claiming the figure describes a real examination process. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
From an exam question to an evidence-safe conclusion
A final result should be numerically sensible and clearly interpreted. The examiner needs to know which event you calculated; a probability without its definition is like a measurement without units. These cases also test whether a mathematical model is incorrectly treated as a measured real-world risk.
Original probability clinic 25: A student says the chance of at least one success over two draws is always 2p.
Find the event. A student says the chance of at least one success over two draws is always 2p. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. The shortcut is invalid when successes can occur on both trials; the overlap must be handled. A complement or disjoint-route calculation is safer. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The arithmetic ‘2p’ can exceed one, proving it cannot be a universal probability rule. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Use p = 0.6 as a counterexample and calculate the independent two-trial complement. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 26: A school draws two distinct tickets from 20, including five winning tickets. At least one winning ticket.
Find the event. A school draws two distinct tickets from 20, including five winning tickets. At least one winning ticket. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No winning ticket means two from 15 losing tickets: 15/20 × 14/19 = 21/38. Therefore at least one winning ticket has probability 17/38. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not say ‘five twentieths twice’ without recognising the changing population. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Write a labelled answer specifying two draws without replacement. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 27: A school draws two tickets from the same 20 but returns the first ticket. At least one winner.
Find the event. A school draws two tickets from the same 20 but returns the first ticket. At least one winner. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. No win = 15/20 × 15/20 = 9/16, so at least one win = 7/16. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The same category counts can produce a different answer because replacement changes the process. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Compare 7/16 with 17/38 and explain their difference. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 28: A teacher gives the result P(no red) = 0.18 for a specified experiment.
Find the event. A teacher gives the result P(no red) = 0.18 for a specified experiment. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. P(at least one red) = 1 − 0.18 = 0.82, if ‘no red’ is exactly the complement in the same sample space. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Do not subtract probabilities from unrelated experiments or differently defined populations. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain why both event definitions must refer to the same experiment. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 29: A question asks for at most one red in two draws.
Find the event. A question asks for at most one red in two draws. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. Its complement is both red. Calculate P(both red) using the appropriate replacement rule, then subtract from one. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. At most one is not the same event as at least one; both can have different probabilities. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Apply the principle to the three-red, two-blue bag and obtain 1 − 3/5 × 2/4 = 7/10. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 30: A probability answer is 1.12 after adding several ordered routes.
Find the event. A probability answer is 1.12 after adding several ordered routes. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. A probability must lie between zero and one. Recount the routes, checking whether any outcome was counted twice or multiplication replaced by addition. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. The total above one is a diagnostic signal, not a plausible final probability. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Explain how drawing the complete outcome tree would reveal overlap. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 31: A report uses the 40-card litter example to claim a 36.4% wildlife injury risk.
Find the event. A report uses the 40-card litter example to claim a 36.4% wildlife injury risk. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. The 71/195 result belongs to drawing at least one line card from a specified deck, not to animal exposure or injury. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. A probability is meaningful only with its sample space, selection rule and event attached. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Write a full sentence giving the correct interpretation of the card result. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
Original probability clinic 32: The mathematics answer is correct but includes no intermediate work.
Find the event. The mathematics answer is correct but includes no intermediate work. Underline the number of trials and any statement about replacement or independence. Then describe the event in words before entering fractions or decimals. If the phrase is ‘at least one’, write its opposite as ‘none’ and check that the two descriptions cover every possible outcome without overlap.
Full reasoning and result. State the complement, write the product of relevant probabilities, subtract from one, and present the simplified result with the event named. In an examination, show the calculation in a clean sequence and keep the route in view. When probabilities are multiplied, explain that the conditions must occur along the stated sequence. When route probabilities are added, verify that the selected complete routes do not overlap. This helps prevent a correct-looking decimal from being an answer to the wrong question.
The error to repair. Correct final numbers may not earn full credit if essential working is omitted where required. Revisit the wording rather than memorising a number. Most persistent probability mistakes come from three sources: the event was mistranslated, the sample space changed unnoticed, or different routes were combined incorrectly. Write one sentence naming the source of the error. That explanation is more useful for your next unseen problem than simply crossing out an answer.
Transfer task. Write the smallest complete set of calculation lines for a three-draw example. Work through it before rereading the model. If you get a different numerical answer, compare the event sets and the counts after each draw. Check that the result lies in the interval from zero to one and that it makes sense relative to the single-trial event. A sound method should still work after the colours or numbers change.
A complete worked demonstration with and without replacement
Return to the bag with three red and two blue counters. In the without-replacement case, draw twice and seek at least one blue. The complement is drawing red and then red: (3/5)(2/4) = 3/10. The answer is 1 − 3/10 = 7/10. Independently, the routes with at least one blue are red–blue, blue–red and blue–blue. Their probabilities are (3/5)(2/4) = 3/10, (2/5)(3/4) = 3/10 and (2/5)(1/4) = 1/10. Add to 7/10. Both approaches agree. The direct route method is longer but particularly useful as a check against a missing outcome.
If the first counter is returned and the bag is mixed before the second draw, the complement is two red selections with replacement: (3/5)(3/5) = 9/25. Therefore at least one blue has probability 1 − 9/25 = 16/25. Now every complete two-draw route has its own probability from the unchanged bag. The change from 7/10 to 16/25 is not a contradiction; it is evidence that the experiment matters. Without replacement, removing a red counter leaves relatively more blue counters in the second draw. With replacement, the starting proportions remain fixed.
How to check work under examination pressure
First, circle the event phrase: ‘at least one’, ‘exactly one’ or ‘at most one’. Second, note the number of draws or trials. Third, underline ‘with replacement’, ‘without replacement’ or an explicit independence assumption. Fourth, name the complement if it is simpler than listing all successful routes. Fifth, write every factor with its correct numerator and denominator. Sixth, calculate, simplify and check the range. Seventh, answer in a sentence that identifies the original event. That is a dependable seven-step method, not seven lines that must appear in every answer.
Students who use a calculator should still understand why the factors are there. A calculator can multiply 32/40 by 31/39 accurately; it cannot tell you whether those factors describe the right event. In Paper 2 particularly, a final application question may combine mathematical information with narrative conditions. The right interpretation of ‘not replaced’, ‘at least one’ and ‘randomly selected’ must come before efficient arithmetic. When stuck, return to a tiny sample space and draw the complete tree to understand the logic.
A week of productive probability revision
Day one: write the sample space for two coin tosses and distinguish three event phrases. Day two: solve two-draw problems with a bag, first with replacement and then without. Day three: use complements for two and three independent trials. Day four: reverse the method by listing every successful ordered route and checking the complement answer. Day five: change the numbers in a previous question and explain why the method still works. Day six: do mixed questions without a chapter heading that gives away the technique. Day seven: revisit only your errors and classify each as event-language, sampling-rule, calculation or interpretation.
A parent or tutor can assist without handing over the formula. Ask: ‘What would make at least one blue false?’ Then ask: ‘What is left in the bag after the first draw?’ If the learner can answer both questions, the fraction usually follows. If not, model with coloured paper counters and let the student physically remove an item. Students may return to the PSLE Learning Guide for fractions and elementary chance before rebuilding the two-stage calculation. That is not a detour; it strengthens the mathematical foundation on which the advanced reasoning depends.
Where to continue
The G3 SEC EMS Learner’s Guide connects examination skills across subjects. Volume 0178 owns probability tree diagrams generally, Volume 0182 explains the difference between exclusive and independent events, and Volume 0258 concentrates on the shifting denominator in without-replacement draws. This chapter adds one distinct competence: identify the complement of ‘at least one’, compute it correctly and interpret the answer without overcounting. Volume 0260’s coastal evidence audit applies a card-draw example while examining claims about environmental surveys, and the Mathematics Learning Guide on dependent and independent events is another subject-owner route.
The principle is small but durable: first say precisely what counts as success. Then ask whether the opposite event is easier. If it is, find its probability from the actual experiment and subtract from one. Use a direct enumeration as a check when helpful. A student who can explain these choices aloud is less likely to lose a mark because a familiar-looking fraction was attached to the wrong event.
