Mathematics

1. 1.

   The values 2, 4, 6 and 8 have frequencies 3, 5, 4 and 2 respectively. Calculate the mean.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Calculates the total frequency as 14. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Calculates the weighted total 2×3 + 4×5 + 6×4 + 8×2. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Obtains the weighted total 66. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: Obtains the mean 66/14 = 4.71 to three significant figures. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Divide the weighted total by the total frequency, not by the number of different values.

   Marking points

   • Calculates the total frequency as 14.
   • Calculates the weighted total 2×3 + 4×5 + 6×4 + 8×2.
   • Obtains the weighted total 66.
   • Obtains the mean 66/14 = 4.71 to three significant figures.

   Examiner tip: Divide the weighted total by the total frequency, not by the number of different values.

2. 2.

   A bag contains 5 red and 3 blue counters. Two counters are taken without replacement. Calculate the probability that both are blue.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Uses probability 3/8 for the first blue counter. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Uses probability 2/7 for the second blue counter. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Multiplies 3/8 by 2/7. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: Obtains 3/28. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Without replacement changes both the numerator and denominator on the second draw.

   Marking points

   • Uses probability 3/8 for the first blue counter.
   • Uses probability 2/7 for the second blue counter.
   • Multiplies 3/8 by 2/7.
   • Obtains 3/28.

   Examiner tip: Without replacement changes both the numerator and denominator on the second draw.

3. 3.

   Find the median and interquartile range of the data set: 3, 7, 8, 12, 14, 15, 19, 22.

   [4 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Identifies the median as the mean of the two middle values (12, 14): median = 13. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Identifies the lower quartile as the median of the lower half (3, 7, 8, 12): Q1 = 7.5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Identifies the upper quartile as the median of the upper half (14, 15, 19, 22): Q3 = 17. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: States the interquartile range = Q3 − Q1 = 9.5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: With an even number of data points, both the median and each quartile are found by averaging the two middle values of the relevant half.

   Marking points

   • Identifies the median as the mean of the two middle values (12, 14): median = 13.
   • Identifies the lower quartile as the median of the lower half (3, 7, 8, 12): Q1 = 7.5.
   • Identifies the upper quartile as the median of the upper half (14, 15, 19, 22): Q3 = 17.
   • States the interquartile range = Q3 − Q1 = 9.5.

   Examiner tip: With an even number of data points, both the median and each quartile are found by averaging the two middle values of the relevant half.

4. 4.

   A fair six-sided die is rolled twice. Calculate the probability that the sum of the two scores is 9.

   [3 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Lists the outcomes summing to 9: (3,6), (4,5), (5,4), (6,3). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: States there are 4 favourable outcomes out of 36 total equally likely outcomes. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Obtains probability = 4/36 = 1/9. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: List outcomes systematically as ordered pairs to avoid missing or double-counting a combination.

   Marking points

   • Lists the outcomes summing to 9: (3,6), (4,5), (5,4), (6,3).
   • States there are 4 favourable outcomes out of 36 total equally likely outcomes.
   • Obtains probability = 4/36 = 1/9.

   Examiner tip: List outcomes systematically as ordered pairs to avoid missing or double-counting a combination.

5. 5.

   A survey of 50 students recorded the number of siblings each has, shown in a frequency table: 0 siblings (12 students), 1 sibling (20), 2 siblings (13), 3 siblings (5). Calculate the mean number of siblings.

   [3 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Calculates the weighted total: (0×12) + (1×20) + (2×13) + (3×5). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Obtains the weighted total = 61. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Divides by 50 to obtain the mean = 1.22 siblings. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Multiply each value by its frequency before summing — do not simply average the raw values 0, 1, 2 and 3.

   Marking points

   • Calculates the weighted total: (0×12) + (1×20) + (2×13) + (3×5).
   • Obtains the weighted total = 61.
   • Divides by 50 to obtain the mean = 1.22 siblings.

   Examiner tip: Multiply each value by its frequency before summing — do not simply average the raw values 0, 1, 2 and 3.

6. 6.

   Two events A and B are independent, with P(A) = 0.3 and P(B) = 0.5. Calculate P(A and B) and P(A or B).

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Uses P(A and B) = P(A) × P(B) for independent events. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Obtains P(A and B) = 0.15. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Uses P(A or B) = P(A) + P(B) − P(A and B). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: Obtains P(A or B) = 0.3 + 0.5 − 0.15 = 0.65. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The subtraction of P(A and B) in the 'or' formula avoids double-counting the overlap between the two events.

   Marking points

   • Uses P(A and B) = P(A) × P(B) for independent events.
   • Obtains P(A and B) = 0.15.
   • Uses P(A or B) = P(A) + P(B) − P(A and B).
   • Obtains P(A or B) = 0.3 + 0.5 − 0.15 = 0.65.

   Examiner tip: The subtraction of P(A and B) in the 'or' formula avoids double-counting the overlap between the two events.

7. 7.

   A cumulative frequency curve for 80 students' test scores shows that the median score is 62 and the 90th percentile is 85. Interpret what the 90th percentile value tells you about the distribution of scores.

   [2 marks] · no calculator

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
   2. Work through this mathematical step: States that 90% of the students scored 85 or below (equivalently, 10% scored above 85). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: States that this indicates the top 10% of students achieved relatively high scores compared to the rest of the group. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: A percentile always describes the percentage of the data falling below (or at) that value, not above it.

   Marking points

   • States that 90% of the students scored 85 or below (equivalently, 10% scored above 85).
   • States that this indicates the top 10% of students achieved relatively high scores compared to the rest of the group.

   Examiner tip: A percentile always describes the percentage of the data falling below (or at) that value, not above it.

8. 8.

   A box contains 4 red pens and 6 blue pens. A pen is chosen at random, its colour noted, and it is replaced before a second pen is chosen. Draw a tree diagram outcome summary and calculate the probability that both pens are the same colour.

   [4 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: States P(red) = 4/10 = 0.4 and P(blue) = 6/10 = 0.6, unchanged for the second draw since the pen is replaced. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Calculates P(both red) = 0.4 × 0.4 = 0.16. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Calculates P(both blue) = 0.6 × 0.6 = 0.36. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Work through this mathematical step: Adds the two mutually exclusive outcomes: P(same colour) = 0.16 + 0.36 = 0.52. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Because the pen is replaced, the two draws are independent, so the probabilities on the second branch of the tree diagram stay the same as the first.

   Marking points

   • States P(red) = 4/10 = 0.4 and P(blue) = 6/10 = 0.6, unchanged for the second draw since the pen is replaced.
   • Calculates P(both red) = 0.4 × 0.4 = 0.16.
   • Calculates P(both blue) = 0.6 × 0.6 = 0.36.
   • Adds the two mutually exclusive outcomes: P(same colour) = 0.16 + 0.36 = 0.52.

   Examiner tip: Because the pen is replaced, the two draws are independent, so the probabilities on the second branch of the tree diagram stay the same as the first.

9. 9.

   A set of 6 numbers has mean 10. A seventh number, 24, is added to the set. Calculate the new mean.

   [3 marks]

   Answer explanation

   Draft walkthroughs are based on marking guidance, not independently verified derivations.

   1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
   2. Work through this mathematical step: Calculates the original total: 6 × 10 = 60. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   3. Work through this mathematical step: Adds the new number: 60 + 24 = 84. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   4. Work through this mathematical step: Divides by the new count of 7 to obtain a new mean of 12. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
   5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Always work back to the total (sum) first when a data set changes — the mean itself cannot be adjusted directly without it.

   Marking points

   • Calculates the original total: 6 × 10 = 60.
   • Adds the new number: 60 + 24 = 84.
   • Divides by the new count of 7 to obtain a new mean of 12.

   Examiner tip: Always work back to the total (sum) first when a data set changes — the mean itself cannot be adjusted directly without it.

10. 10.

    Explain why the mode is a more appropriate average than the mean for a data set that includes categorical data such as favourite colour.

    [2 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: States that the mean requires numerical values that can be added and divided, which categorical data (like colours) does not have. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that the mode simply identifies the most frequently occurring category, which is meaningful and calculable for non-numerical data. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The choice of average depends on the type of data: mode works for any data type, median needs ordered data, and mean needs numerical data.

    Marking points

    • States that the mean requires numerical values that can be added and divided, which categorical data (like colours) does not have.
    • States that the mode simply identifies the most frequently occurring category, which is meaningful and calculable for non-numerical data.

    Examiner tip: The choice of average depends on the type of data: mode works for any data type, median needs ordered data, and mean needs numerical data.

11. 11.

    The table shows the time (in minutes) taken by 40 students to complete a puzzle: 0 < t ≤ 10 (8 students), 10 < t ≤ 20 (15), 20 < t ≤ 30 (12), 30 < t ≤ 40 (5). Calculate an estimate of the mean time, using the midpoint of each interval.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Identifies the midpoints of each interval as 5, 15, 25 and 35. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Calculates the weighted total: (5×8) + (15×15) + (25×12) + (35×5). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Obtains the weighted total = 740. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: Divides by 40 to obtain an estimated mean of 18.5 minutes. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Using the midpoint of each class is only ever an estimate of the mean, since the exact individual values within each group are unknown.

    Marking points

    • Identifies the midpoints of each interval as 5, 15, 25 and 35.
    • Calculates the weighted total: (5×8) + (15×15) + (25×12) + (35×5).
    • Obtains the weighted total = 740.
    • Divides by 40 to obtain an estimated mean of 18.5 minutes.

    Examiner tip: Using the midpoint of each class is only ever an estimate of the mean, since the exact individual values within each group are unknown.

12. 12.

    A histogram has a bar covering the class interval 10 to 25 (a width of 15), with a frequency density of 2.4. Calculate the frequency for this class interval.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the formula: frequency = frequency density × class width. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Substitutes 2.4 × 15. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the frequency as 36. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: In a histogram, it is the area of each bar (frequency density × width) that represents frequency, not the height alone — this matters whenever class widths are unequal.

    Marking points

    • States the formula: frequency = frequency density × class width.
    • Substitutes 2.4 × 15.
    • States the frequency as 36.

    Examiner tip: In a histogram, it is the area of each bar (frequency density × width) that represents frequency, not the height alone — this matters whenever class widths are unequal.

13. 13.

    In a class of 30 students, 18 study French, 15 study Spanish, and 8 study both. Calculate the number of students who study neither language.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Uses n(French or Spanish) = n(French) + n(Spanish) − n(both) = 18 + 15 − 8. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Obtains n(French or Spanish) = 25. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the number studying neither as 30 − 25 = 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Subtracting the overlap once (not twice) is the key step whenever two groups share common members — a Venn diagram makes this easy to see visually.

    Marking points

    • Uses n(French or Spanish) = n(French) + n(Spanish) − n(both) = 18 + 15 − 8.
    • Obtains n(French or Spanish) = 25.
    • States the number studying neither as 30 − 25 = 5.

    Examiner tip: Subtracting the overlap once (not twice) is the key step whenever two groups share common members — a Venn diagram makes this easy to see visually.

14. 14.

    Using the class in the previous question (18 study French, 15 study Spanish, 8 study both), a student is selected at random from those who study French. Calculate the probability that this student also studies Spanish.

    [2 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the probability as the number studying both divided by the number studying French: 8/18. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Simplifies to 4/9. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Restricting the random selection to a named subgroup (here, French students) is exactly what makes this a conditional probability, with that subgroup's size as the denominator.

    Marking points

    • States the probability as the number studying both divided by the number studying French: 8/18.
    • Simplifies to 4/9.

    Examiner tip: Restricting the random selection to a named subgroup (here, French students) is exactly what makes this a conditional probability, with that subgroup's size as the denominator.

15. 15.

    A biased coin is thrown 200 times and lands on heads 140 times. (a) Calculate the relative frequency of heads. (b) State the best estimate of the probability that the next throw lands on heads.

    [2 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the relative frequency as 140/200 = 0.7. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that the best estimate of the probability is also 0.7, using the relative frequency from a large number of trials. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For a biased object, relative frequency from many trials is the only way to estimate a probability — it cannot be assumed equally likely like a fair coin.

    Marking points

    • Calculates the relative frequency as 140/200 = 0.7.
    • States that the best estimate of the probability is also 0.7, using the relative frequency from a large number of trials.

    Examiner tip: For a biased object, relative frequency from many trials is the only way to estimate a probability — it cannot be assumed equally likely like a fair coin.

16. 16.

    A cumulative frequency table for the number of books read by 60 students in a year shows: ≤5 books (10 students), ≤10 books (28), ≤15 books (48), ≤20 books (60). Use linear interpolation to estimate the median number of books read.

    [4 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: Identifies the median position as the 30th value (60 ÷ 2). Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Identifies that the 30th value falls within the 10-15 books class, since cumulative frequency rises from 28 to 48 across it. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Uses linear interpolation: median ≈ 10 + [(30 − 28)/20] × 5. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Work through this mathematical step: States the estimated median as 10.5 books. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    6. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Linear interpolation assumes the values are evenly spread within a class — it gives a sensible estimate from grouped data, not an exact value.

    Marking points

    • Identifies the median position as the 30th value (60 ÷ 2).
    • Identifies that the 30th value falls within the 10-15 books class, since cumulative frequency rises from 28 to 48 across it.
    • Uses linear interpolation: median ≈ 10 + [(30 − 28)/20] × 5.
    • States the estimated median as 10.5 books.

    Examiner tip: Linear interpolation assumes the values are evenly spread within a class — it gives a sensible estimate from grouped data, not an exact value.

17. 17.

    A survey of 60 people recorded whether they like a sport and their gender: 24 males like it, 6 males dislike it, 18 females like it, 12 females dislike it. Calculate the probability that a randomly selected person from the survey is female and likes the sport.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the total number surveyed: 24 + 6 + 18 + 12 = 60. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Identifies the number who are female and like the sport as 18. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: States the probability as 18/60 = 3/10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: For a combined condition ('and'), read the single cell of the two-way table where both conditions meet, then divide by the grand total.

    Marking points

    • Calculates the total number surveyed: 24 + 6 + 18 + 12 = 60.
    • Identifies the number who are female and like the sport as 18.
    • States the probability as 18/60 = 3/10.

    Examiner tip: For a combined condition ('and'), read the single cell of the two-way table where both conditions meet, then divide by the grand total.

18. 18.

    A bag contains 4 green and 6 yellow balls. Three balls are drawn at random without replacement. Calculate the probability that all three balls are yellow.

    [3 marks]

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: States the probability of the first ball being yellow as 6/10. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States the probabilities for the second and third draws as 5/9 and 4/8, adjusting the total each time since there is no replacement. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Work through this mathematical step: Multiplies and simplifies 6/10 × 5/9 × 4/8 to obtain 1/6. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    5. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Without replacement, both the number of favourable outcomes and the total number remaining decrease by one after each draw.

    Marking points

    • States the probability of the first ball being yellow as 6/10.
    • States the probabilities for the second and third draws as 5/9 and 4/8, adjusting the total each time since there is no replacement.
    • Multiplies and simplifies 6/10 × 5/9 × 4/8 to obtain 1/6.

    Examiner tip: Without replacement, both the number of favourable outcomes and the total number remaining decrease by one after each draw.

19. 19.

    A scatter diagram shows a strong positive correlation between hours studied and exam score, based on data for students who studied between 2 and 10 hours. State what a strong positive correlation means, and explain one reason it would be unreliable to use the line of best fit to predict the score for a student who studied 25 hours.

    [2 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. Break the command into its requested parts. For each part, connect a relevant fact or observation to the conclusion it supports. Describing what happens and explaining why it happens are different tasks.
    2. Work through this mathematical step: States that a strong positive correlation means that as one variable (hours studied) increases, the other variable (exam score) also tends to increase, closely following a linear pattern. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: Explains that 25 hours lies far outside the original data range (2 to 10 hours), so extrapolating this far is unreliable since the relationship may not continue in the same way beyond the observed data. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: Using a line of best fit to predict far beyond the range of the original data (extrapolation) is always less reliable than predicting within that range (interpolation).

    Marking points

    • States that a strong positive correlation means that as one variable (hours studied) increases, the other variable (exam score) also tends to increase, closely following a linear pattern.
    • Explains that 25 hours lies far outside the original data range (2 to 10 hours), so extrapolating this far is unreliable since the relationship may not continue in the same way beyond the observed data.

    Examiner tip: Using a line of best fit to predict far beyond the range of the original data (extrapolation) is always less reliable than predicting within that range (interpolation).

20. 20.

    A box-and-whisker plot has minimum 4, lower quartile 10, median 15, upper quartile 22, and maximum 30. Calculate the interquartile range, and state whether a new value of 35 could belong to this same data set, giving a reason.

    [2 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. List the given quantities and the requested unknown. Choose the relation that connects them, state any required assumptions, then substitute before rounding. Preserve exact expressions when the task asks for an exact result.
    2. Work through this mathematical step: Calculates the interquartile range as upper quartile − lower quartile = 22 − 10 = 12. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    3. Work through this mathematical step: States that 35 could not belong to this recorded data set, because it exceeds the given maximum value of 30. Write the intermediate operation, keep the units consistent where applicable, and check the relation against the quantities given in the question.
    4. Check the complete task again, including restrictions, units, precision and supporting evidence when relevant. Specific caution: The maximum and minimum on a box plot are the actual highest and lowest recorded values in the data set — no value in that data set can lie outside them.

    Marking points

    • Calculates the interquartile range as upper quartile − lower quartile = 22 − 10 = 12.
    • States that 35 could not belong to this recorded data set, because it exceeds the given maximum value of 30.

    Examiner tip: The maximum and minimum on a box plot are the actual highest and lowest recorded values in the data set — no value in that data set can lie outside them.

21. 21.

    A bag contains three red and two blue counters. Two are drawn without replacement. Find the probability that both are red.

    [3 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. The second draw depends on the first because the chosen counter is not returned.
    2. Follow the red-red path: both the red count and total count decrease after the first draw.

    Marking points

    • Probability of red first = 3/5.
    • After a red, probability of red second = 2/4.
    • Multiply: (3/5)(2/4) = 3/10.

    Examiner tip: Using 3/5 for both draws incorrectly assumes replacement.

22. 22.

    A bag contains four red and three blue counters. Two are drawn without replacement. Given that at least one is blue, find the probability that both are blue.

    [5 marks] · no calculator

    Answer explanation

    Draft walkthroughs are based on marking guidance, not independently verified derivations.

    1. The condition removes only red-red outcomes, leaving a smaller sample space.
    2. Among the equally likely unordered pairs there are 21 total, 6 red-red and 3 blue-blue, giving 3/(21 - 6) = 1/5 as an independent check.

    Marking points

    • P(both blue) = (3/7)(2/6) = 1/7.
    • P(both red) = (4/7)(3/6) = 2/7.
    • P(at least one blue) = 1 - 2/7 = 5/7.
    • Both blue is a subset of at least one blue, so divide the two probabilities.
    • Conditional probability = (1/7)/(5/7) = 1/5.

    Examiner tip: At least one blue includes both blue; it does not mean exactly one blue.

Original, unofficial practice questions from SubjectScout — not official past-paper questions.