Statistics and Probability — Free IB MYP4 Mathematics (Standard) Questions

1QuestionFairness Bias and RandomnessConcept Practice

4 marks~6 minCriterion A

A quality-control technician suspects a six-sided die used in a board-game factory may be biased. The die is rolled 60 times with the following results.

Outcome	1	2	3	4	5	6
Frequency	8	10	9	7	13	13

a

Calculate the relative frequency of rolling a 5. [1]

b

Calculate the absolute difference between the relative frequency of rolling a 5 and its theoretical probability. [2]

c

The factory rejects any die for which the absolute difference between the relative frequency and theoretical probability of any outcome exceeds 0.08 after 60 trials. Justify whether this die should be rejected, supporting your answer with a numerical comparison. [1]

Solutions

2QuestionUsing Probability Trees for Multi-Stage EventsConcept Practice

2 marks~3 minCriterion A

A quality-control engineer tests two components in sequence from a large batch. Long-run data show the probability that any single component is defective is

0.2

, independent of all other components. A probability tree shows the two-stage process with branches labelled D (defective) and N (non-defective).

a

Interpret the meaning of the branch probability

P(\text{D}) = 0.2

at the first stage, and use the multiplication rule to calculate

P(\text{ND})

, showing your substitution explicitly. [1]

b

Explain why the probabilities on the branches leaving any single node must sum to 1, and state what this implies about the outcomes at that node. [1]

Solutions

3QuestionStem and Leaf PlotsConcept Practice

3 marks~5 minCriterion A

A school uses a minimum median score of 72 to determine whether a class needs additional support. The stem-and-leaf plot below shows the test scores (out of 100) of 15 students.

\begin{array}{r|l}\textbf{Stem} & \textbf{Leaf} \\\hline4 & 5 \\5 & 2 \quad 8 \\6 & 1 \quad 4 \quad 7 \\7 & 0 \quad 3 \quad 5 \quad 6 \\8 & 2 \quad 4 \quad 8 \quad 9 \\9 & 1 \\\end{array}

Key:

4 \mid 5

represents a score of 45.

a

Calculate the median test score. [1]

b

Deduce the position of the median in this data set, showing the method used to identify it. [1]

c

Justify whether this class requires additional support, with reference to the school's criterion of 72. [1]

Solutions

4QuestionConstructing and Interpreting Box PlotsConcept Practice

2 marks~3 minCriterion A

A school uses box plots to compare class performance on a standardised mathematics test.

The five-number summary for Class 10B is:
Minimum

= 40

,

Q_1 = 55

, Median

= 70

,

Q_3 = 85

, Maximum

= 100

.

The school's policy states that a class shows sufficient spread in the middle 50% of scores only if the IQR exceeds 25 marks.

a

Calculate the interquartile range (IQR) for Class 10B. [1]

b

Justify whether Class 10B meets the school's policy for sufficient spread. [1]

Solutions

5QuestionComparing Experimental and Expected ResultsAssessment Practice

2 marks~3 minCriterion D

A factory produces 5000 light bulbs each day. The manufacturer claims that 2% of bulbs are defective. A quality inspector tests a random sample of 50 bulbs and finds 4 defective bulbs. The expected number of defective bulbs in the sample is

2\% \times 50 = 1

.

a

Describe a real-world context in which experimental results are compared to theoretical expectations, using this scenario as an example. [1]

b

Advise the quality inspector whether the sample of 50 bulbs is sufficient to conclude that the factory's defect rate has genuinely increased beyond the manufacturer's 2% claim. [1]

Solutions

6QuestionRelative Frequency and Long-Term TrendsAssessment Practice

6 marks~9 minCriterion C

A quality-control analyst at a board-game manufacturer tests whether a six-sided die is fair. She rolls the die 60 times and records that the face showing 4 appears 8 times.

a

Calculate the relative frequency of rolling a 4 from these 60 trials. [1]

b

Deduce the expected number of times a 4 should appear in 60 rolls of a fair die, and explain what the difference between this value and the observed count reveals about experimental and theoretical probability. [2]

c

The analyst concludes: "60 rolls is not enough evidence to declare the die unfair; I need far more trials." Justify whether her conclusion is mathematically sound, using the law of large numbers to support your reasoning. [3]

Solutions

7QuestionApplications in Real-World Risk AssessmentAssessment Practice

4 marks~6 minCriterion D

An insurance company records the following data for two groups of drivers over one year.

Age group	Under 25 — claims: 120	total drivers: 800
Age group	Over 25 — claims: 80	total drivers: 1200

The relative risk of making a claim is defined as

\text{Relative Risk} = \frac{P(\text{claim} \mid \text{under 25})}{P(\text{claim} \mid \text{over 25})}

a

Calculate

P(\text{claim} \mid \text{under 25})

and

P(\text{claim} \mid \text{over 25})

. [2]

b

Calculate the relative risk of making a claim for drivers under 25 compared to drivers over 25. [1]

c

The insurance company charges drivers under 25 a premium 2.5 times higher than drivers over 25. Justify whether this premium multiplier is supported by the claim data. [1]

Solutions

8QuestionDesigning Surveys and QuestionnairesAssessment Practice

6 marks~9 minCriterion B

A school surveys students about their lunch preferences. Responses are recorded across four 2-hour time slots over three consecutive days.

Time slots: 8–10, 10–12, 12–14, 14–16

Day 1	24	48	36	18
Day 2	30	60	45	22
Day 3	36	72	54	26

a

Analyse the relationship between the 8–10 and 10–12 response counts, and between the 10–12 and 12–14 response counts, for each day. [2]

b

Deduce the predicted number of responses for the 10–12 slot on Day 4. Show your working. [2]

c

The school will only adjust the lunch menu if the 10–12 slot consistently attracts more than 1.5 times the responses of any other single slot across all days. Advise the school whether the lunch menu should be adjusted, justifying your answer using your findings from parts (a) and (b). [2]

Solutions

9QuestionTypes of Data Discrete vs ContinuousAssessment Practice

6 marks~9 minCriterion A

A student is designing a survey to investigate a possible relationship between the number of hours spent on social media per day and the number of books read per month. She has formulated eight survey questions:

1. How many hours do you spend on social media each day? (Choose one: 0–1, 1–2, 2–3, 3–4, 4+)
2. What is your favourite social media platform?
3. How many books have you read in the past month?
4. What is your age in years?
5. How many minutes did you spend on social media yesterday?
6. What is your favourite colour?
7. How many siblings do you have?
8. What is the temperature in your room in degrees Celsius?

a

Classify each of the eight questions as collecting either discrete or continuous data. Explain your reasoning for one question of each type. [2]

b

Construct three new survey questions that collect discrete data relevant to the same investigation. For each question, specify the potential data values it would produce and explain why those values are discrete. [2]

c

Construct three new survey questions that collect continuous data relevant to the same investigation. For each question, specify the potential data values it would produce. Justify whether the continuous data from your questions would be more useful to this investigation than the discrete data from part (b). [2]

Solutions

10QuestionTypes of Data Discrete vs ContinuousAssessment Practice

8 marks~12 minCriterion D

A hospital records four types of patient data: heart rate (beats per minute, to one decimal place), number of hospital visits in the past year, blood type (A, B, AB, O), and body temperature (degrees Celsius, to one decimal place).

a

Classify each of the four data types as discrete or continuous. Justify each classification. [2]

b

A software bug rounds all continuous data to the nearest integer before storing it. A patient's true heart rate fluctuates between

71.2

bpm and

72.8

bpm, but is stored as

72

bpm throughout. Discuss how this rounding could affect the detection of arrhythmias (irregular heartbeats). [3]

c

Advise the hospital's data team whether the integer-storage model is suitable for recording continuous medical data, addressing diagnostic accuracy, patient safety, and the limitations this places on the data model. [3]

Solutions

11QuestionDesigning Surveys and QuestionnairesAssessment Practice

2 marks~3 minCriterion C

A school survey asks: "Do you agree that starting school later would improve student well-being and academic performance?"

a

Explain why the wording of this question may produce biased responses. [1]

b

Propose a revised survey question that minimises bias, and justify how your revision better reflects the full range of student opinions. [1]

Solutions

12QuestionRecognizing and Modeling Dependent EventsAssessment Practice

5 marks~8 minCriterion B

A quality-control technician samples marbles from a production batch. Each bag contains exactly 3 blue marbles and some red marbles, with

n

marbles in total. Two marbles are drawn in succession without replacement.

n

: 3, 4, 5, 6

P(\text{1st blue})

:

\dfrac{3}{3}

,

\dfrac{3}{4}

,

\dfrac{3}{5}

,

\dfrac{3}{6}

P(\text{2nd blue} \mid \text{1st blue})

:

\dfrac{2}{2}

,

\dfrac{2}{3}

,

\dfrac{2}{4}

,

\dfrac{2}{5}

P(\text{both blue})

:

1

,

\dfrac{1}{2}

,

\dfrac{3}{10}

,

\dfrac{1}{5}

a

Deduce a general formula for

P(\text{both blue})

in terms of

n

. [2]

b

Calculate

P(\text{both blue})

for a bag of 10 marbles. [1]

c

The technician rejects any batch where the probability of drawing two blue marbles in succession is less than

\dfrac{1}{20}

. Justify whether a bag of 12 marbles should be rejected, and advise the technician on what this threshold indicates about the production process. [2]

Solutions

13QuestionUsing Probability Trees for Multi-Stage EventsAssessment Practice

7 marks~11 minCriterion D

A quality-control engineer tests microchips by running them through a sequence of independent diagnostic checks. At each check, a chip either passes (P) or fails (F), each with probability

\dfrac{1}{2}

.

a

Write down the number of distinct outcomes after 1 check, after 2 checks, and after 3 checks. Identify the pattern in your sequence. [2]

b

Deduce a formula for the total number of distinct outcomes after

n

checks, each with

m

equally likely results. Justify your formula using the branching structure of a probability tree. [2]

c

The engineer runs 3 checks on each chip. Construct a probability tree for this 3-stage process. A chip is accepted only if it passes at least 2 of the 3 checks. Calculate the probability that a chip is accepted, and evaluate whether this acceptance criterion is suitable for a production line. [3]

Solutions

14QuestionReal-Life Applications Games Genetics and Decision-MakingAssessment Practice

2 marks~3 minCriterion C

A genetic cross of pea plants produces 60 offspring. Their phenotypes are recorded below.

Smooth texture, Yellow colour: 30
Smooth texture, Green colour: 15
Wrinkled texture, Yellow colour: 10
Wrinkled texture, Green colour: 5

a

Calculate

P(\text{Yellow} \mid \text{Smooth})

, the conditional probability that a randomly selected smooth plant is yellow. Show your working. [1]

b

The overall proportion of yellow plants in the dataset is

\dfrac{40}{60} = \dfrac{2}{3}

. Justify whether the botanist's claim — that knowing a plant is smooth gives useful information about predicting its colour — is supported by the data. [1]

Solutions

15QuestionComparing Data Sets Using AveragesAssessment Practice

6 marks~9 minCriterion A

A city's environmental agency monitors daily air quality scores over five days.

Set A (recorded scores):

5,\ 8,\ 12,\ 15,\ 20

Due to a calibration improvement, every score in a revised data set, Set B, is exactly

k

points higher than the corresponding score in Set A, giving values

5+k,\ 8+k,\ 12+k,\ 15+k,\ 20+k

.

a

Calculate the mean air quality score for Set A.

[2]

b

Show that the mean of Set B equals

12 + k

.

[2]

c

The agency considers air quality "satisfactory" if the mean score is at least

15

. The minimum integer value of

k

for which Set B meets this standard is

k = 3

. Justify whether this calibration adjustment is sufficient to conclude that the original instrument was under-reading by exactly

3

units.

[2]

Solutions

16QuestionReal-Life Interpretation of AveragesAssessment Practice

6 marks~9 minCriterion B

A technology company records the following salary data for its employees.

Salary (USD)	30 000	40 000	50 000	60 000	70 000
Number of employees	3	7	12	5	3

A second company advertises a mean salary of 50 000 USD to attract new hires. Its salary data are:

Salary (USD)	35 000	45 000	55 000	65 000
Number of employees	4	6	8	2

a

Calculate the mean salary for the first company. [2]

b

Deduce the general formula for the weighted mean

\bar{x}

of a dataset where each value

x_i

occurs with frequency

f_i

, using the structure of your working in part (a) to guide your reasoning. [2]

c

Using your formula from part (b), calculate the mean salary for the second company, then advise a prospective employee whether the advertised figure of 50 000 USD gives a reliable picture of typical earnings at that company. Justify your answer. [2]

Solutions

17QuestionMean from a Frequency TableAssessment Practice

6 marks~9 minCriterion C

A city planner is analysing household income data from a small town to decide where to locate a new community centre. The data are summarised below.

Income range (thousand dollars per year)	10–30	30–50	50–70	70–90	90–110	110–130	130–150
Number of households	45	62	38	20	12	8	5

a

Estimate the mean annual household income. Use the midpoint of each interval as the representative value. [2]

b

Evaluate whether the mean is an appropriate measure of central tendency for this dataset. In your response, identify the shape of the distribution and compare the mean with the median as a basis for deciding where to locate the community centre. [2]

c

The planner receives a second dataset for a neighbouring town in which the exact income of every household is recorded. Advise the planner whether to base the location decision on the grouped data from this town or the exact data from the neighbouring town, justifying your recommendation with reference to the reliability of each dataset. [2]

Solutions

18QuestionChoosing the Appropriate MeasureAssessment Practice

2 marks~3 minCriterion D

A hospital records the annual salaries (in thousands of dollars) of its staff. The distribution is right-skewed due to a small number of highly paid specialists.

Explain what the relationship

\text{mean} > \text{median} > \text{mode}

reveals about this salary distribution, and justify which measure of central tendency the hospital should report to best represent a typical staff salary. [2]

Solutions

19QuestionRange and Interquartile RangeAssessment Practice

8 marks~12 minCriterion B

A city's air-quality monitoring network records daily fine-particle counts (µg/m³) at five sensors. On days following an industrial emission event, the highest reading rises progressively while the other four remain fixed:

Day 1:

2, 4, 6, 8, 10

Day 2:

2, 4, 6, 8, 100

Day 3:

2, 4, 6, 8, 200

Day 4:

2, 4, 6, 8, 500

Day 5:

2, 4, 6, 8, 1000

a

Calculate the range and interquartile range (IQR) for each of the five datasets. [2]

b

Deduce the range and IQR for the dataset

2, 4, 6, 8, 5000

, justifying your answer using the pattern observed in part (a). [2]

c

Calculate the range and IQR for the dataset

2, 4, 6, 8, 5000

and verify whether your deduction in part (b) is correct. [2]

d

The health authority uses the IQR, not the range, to assess whether air-quality readings are "stable" across sensors. Advise the health authority whether this choice is sufficient for reliable stability assessment, using your results to support your recommendation. [2]

Solutions

20QuestionComparing Variability Between Data SetsAssessment Practice

8 marks~12 minCriterion D

A school awards a scholarship to the candidate who performs most consistently across eight tests, each marked out of 100. Two finalists have the following scores, recorded in the order the tests were taken.

Candidate X: 85, 88, 82, 90, 86, 84, 89, 84

Candidate Y: 95, 75, 98, 72, 94, 78, 92, 76

a

Calculate the mean and population standard deviation for each candidate. Show all working. [4]

b

Deduce which candidate demonstrates greater consistency. Justify your answer using both measures calculated in part (a). [2]

c

Advise the scholarship panel whether standard deviation alone provides a sufficient basis for their decision, given that the scores are recorded in the order the tests were taken. [2]

Solutions

21QuestionComparing Variability Between Data SetsAssessment Practice

3 marks~5 minCriterion C

A mathematics coordinator is reviewing test performance across two classes to decide whether to report variability using the range or the interquartile range (IQR) in the end-of-year report.

The five-number summaries are:

Class A: Minimum

= 50

,

Q_1 = 65

, Median

= 75

,

Q_3 = 85

, Maximum

= 95

Class B: Minimum

= 40

,

Q_1 = 70

, Median

= 75

,

Q_3 = 80

, Maximum

= 90

a

Calculate the range and IQR for each class. [1]

b

Explain which measure of dispersion — range or IQR — better represents the variability of scores for each class, referring to your values from part (a). [1]

c

The coordinator states: "Both classes show similar variability in student performance." Assess whether this statement is supported or contradicted by the data, referring to an appropriate measure of dispersion. [1]

Solutions

22QuestionCumulative Frequency Graphs and CurvesAssessment Practice

2 marks~3 minCriterion D

A school awards a distinction for test scores of 60 or above. The cumulative frequency curve below shows the scores of 80 students.

a

Explain how to read the median

Q_2

from the cumulative frequency curve, using correct mathematical notation. [1]

b

The interquartile range

\text{IQR} = Q_3 - Q_1

is used to decide whether to review the distinction threshold. Justify whether a large IQR supports or undermines confidence in the 60-point threshold. [1]

Solutions

23QuestionDrawing and Reading HistogramsAssessment Practice

6 marks~9 minCriterion C

A city council presents a histogram of residents' commute times (in minutes) to justify cutting funding for a proposed bus route. The histogram uses unequal bin widths but plots raw frequency as bar height.

Bin (minutes)	0–10	10–30	30–60
Frequency	200	300	150

a

Calculate the frequency density for each bin. [3]

b

Explain why plotting raw frequency as bar height, rather than frequency density, produces a misleading histogram when bin widths are unequal. Refer to your values from part (a) in your answer. [1]

c

Advise the council whether this histogram provides sufficient justification for cutting the bus-route funding. In your response, identify the group of commuters most likely to benefit from the bus route, explain how the visual distortion affects the council's argument, and state one additional piece of data the council should present to support a reliable decision. [2]

Solutions

24QuestionConstructing and Interpreting Box PlotsAssessment Practice

6 marks~9 minCriterion B

A sports analyst records the recovery times (in minutes) of 23 athletes after a training session. The ordered values are:

4, 8, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72

Quartile positions for datasets of size

n = 7, 11, 15, 19

are given below.

n = 7

:

Q_1

at position 2, median at position 4,

Q_3

at position 6

n = 11

:

Q_1

at position 3, median at position 6,

Q_3

at position 9

n = 15

:

Q_1

at position 4, median at position 8,

Q_3

at position 12

n = 19

:

Q_1

at position 5, median at position 10,

Q_3

at position 15

a

Deduce a general rule, in terms of

n

, for the positions of the median,

Q_1

, and

Q_3

in an ordered dataset of

n

values. [2]

b

Using your rules from part (a), construct the five-number summary for the 23 recovery times, showing all position calculations. [2]

c

The analyst claims that the middle 50% of athletes recover within a 36-minute window. Justify whether the interquartile range supports this claim. [2]

Solutions

25QuestionCalculating Combined ProbabilitiesAssessment Practice

8 marks~12 minCriterion B

A quality-control technician at a marble factory tests batches by drawing two marbles without replacement from a bag containing equal numbers of red and blue marbles.

Experiment 1: 2 red, 2 blue marbles
Experiment 2: 3 red, 3 blue marbles
Experiment 3: 4 red, 4 blue marbles

a

Calculate the probability of drawing a red marble first and a blue marble second for each experiment. Write each answer as a simplified fraction. [3]

b

Deduce a formula for this probability when the bag contains

n

red and

n

blue marbles, where

n

is a positive integer. [2]

c

The factory considers a batch "well-mixed" if the probability of drawing one marble of each colour (in either order) exceeds

\dfrac{1}{2}

. Advise the factory whether this standard is a useful quality-control threshold, justifying your answer algebraically. [3]

Solutions

26QuestionTree Diagrams for Dependent EventsAssessment Practice

6 marks~9 minCriterion A

An inspection tray contains 5 conforming components and 3 defective components. Two components are selected at random without replacement.

a

Construct a tree diagram showing all possible outcomes of the two selections, with the probability of each branch clearly labelled. [3]

b

Deduce the probability that both components selected are conforming. [1]

c

A batch fails inspection if the probability of selecting at least one defective component exceeds

\dfrac{1}{2}

. Calculate the probability of selecting at least one defective component and justify whether this batch passes or fails inspection. [2]

Solutions

27QuestionSolving Problems using tree diagrams and Venn DiagramsAssessment Practice

4 marks~6 minCriterion C

A bag contains 4 red marbles and 6 blue marbles. Two marbles are drawn without replacement. The partially completed tree diagram shows the first and second draws.

First draw:

P(\text{Red}) = \dfrac{4}{10}

,

P(\text{Blue}) = \dfrac{6}{10}

Second draw (after Red):

P(\text{Red} \mid \text{Red}) = \dfrac{3}{9}

,

P(\text{Blue} \mid \text{Red}) = \dfrac{6}{9}

Second draw (after Blue):

P(\text{Red} \mid \text{Blue}) = \dfrac{4}{9}

,

P(\text{Blue} \mid \text{Blue}) = \dfrac{5}{9}

a

State

P(\text{Red} \mid \text{Blue})

, the probability of drawing a red marble second given that the first marble drawn was blue. [1]

b

Calculate the probability that exactly one red marble is drawn in the two draws. [2]

c

A quality-control inspector claims that, for a fair sampling process, the colour drawn first should have no influence on subsequent draws. Justify whether this bag of marbles satisfies that claim when sampling without replacement. [1]

Solutions

28QuestionComplementary Events and Basic RulesAssessment Practice

6 marks~9 minCriterion D

A hospital uses a rapid test for a rare disease. The test has a sensitivity of 95% (probability of testing positive given the disease is present) and a specificity of 90% (probability of testing negative given the disease is absent). The disease affects 2% of the population.

Let

D

= has the disease and

T^+

= tests positive.

a

Using the complementary rule and the law of total probability, calculate

P(T^+)

, the probability that a randomly selected person tests positive. [2]

b

Calculate

P(D \mid T^+)

, the probability that a person who tests positive actually has the disease. [2]

c

A public health official proposes rolling out this test as a mass screening programme. Evaluate whether this proposal is justified, using your results from (a) and (b) to support your judgement. [2]

Solutions

Statistics and Probability — Free MYP4 Mathematics (Standard) Practice Questions