Statistics and Probability — Free IB MYP5 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 4. [1]

b

Calculate the absolute difference between the relative frequency of rolling a 4 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. Advise the technician whether this die should be rejected, justifying your answer with a numerical comparison. [1]

Solutions

2QuestionDesigning Surveys and QuestionnairesConcept Practice

2 marks~3 minCriterion A

A school has

N

students listed on its register.

Describe a method a student could use to select a simple random sample of 100 students from this register. [2]

Solutions

3QuestionCalculating Mean Median and ModeConcept Practice

2 marks~3 minCriterion A

A meteorologist records the daily maximum temperature (°C) in a coastal city over

n

days, labelling each value

x_1, x_2, x_3, \ldots, x_n

.

State the formula for the arithmetic mean

\bar{x}

of these

n

values, using summation notation. [2]

Solutions

4QuestionBox and Whisker Plots for Spread of DataConcept Practice

4 marks~6 minCriterion A

A school's sports coordinator records the sprint times (in seconds) of 30 students during a fitness assessment. The box-and-whisker plot displays the results, with the following five-number summary:

Minimum

= 8.4

s,

Q_1 = 10.2

s, Median

= 11.8

s,

Q_3 = 13.6

s, Maximum

= 16.0

s

Students who score within 4 seconds of the median are classified as "consistent performers."

a

Calculate the range of the sprint times. [1]

b

Calculate the interquartile range (IQR) of the sprint times. [1]

c

Justify whether the middle 50% of students qualify as consistent performers, using the IQR. [2]

Solutions

5QuestionSolving Problems using tree diagrams and Venn DiagramsConcept Practice

6 marks~9 minCriterion A

A school survey of 100 students recorded participation in soccer (event

A

) and basketball (event

B

). The results are summarised below:

- Students playing only soccer: 30
- Students playing only basketball: 25
- Students playing both sports: 15
- Students playing neither sport: 30

a

State the value of

n(A \cap B)

. [1]

b

Calculate

n(A)

and

n(B)

. [2]

c

Using the formula

n(A \cup B) = n(A) + n(B) - n(A \cap B)

, calculate

n(A \cup B)

. [1]

d

The school requires that more than 65% of surveyed students participate in at least one sport to qualify for additional funding. Justify whether this school meets the requirement. [2]

Solutions

6QuestionApplications 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

7QuestionComparing Experimental and Expected ResultsAssessment Practice

3 marks~5 minCriterion C

A fair coin is flipped 50 times, producing 28 heads and 22 tails.

a

Calculate the experimental relative frequency of heads. [1]

b

Deduce how many heads must occur in the next 50 flips so that the overall relative frequency across all 100 flips equals the theoretical probability of 0.5. [1]

c

Justify whether achieving exactly this number of heads in the next 50 flips would confirm that the coin is fair. [1]

Solutions

8QuestionApplications in Real-World Risk AssessmentAssessment Practice

2 marks~3 minCriterion D

An insurance company estimates that the probability of a house fire in a given region is

5\%

, based on historical data across thousands of properties. They apply this figure to assess the fire risk for an individual house valued at

200\,000

dollars.

Explain one limitation of using this regional probability to assess the fire risk for a specific house, and identify one individual factor that could cause the actual risk to differ from

5\%

. [2]

Solutions

9QuestionDesigning 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

10QuestionDesigning Surveys and QuestionnairesAssessment Practice

6 marks~9 minCriterion C

A school wants to estimate the average number of hours students spend studying each week. During exam week, the survey team asks every student present in the library. These 80 students report an average of 18 hours per week. The remaining 720 students, surveyed separately, report an average of 6 hours per week.

a

Identify the type of sampling bias present in this survey design and explain how it causes the library-only result to overestimate the true school-wide average. [2]

b

Calculate the true average weekly study hours for all 800 students. Show your working clearly. [2]

c

The school principal states: "An average of 7.2 hours per week is too low to reflect real student effort — we should use the library figure of 18 hours when reporting to parents."

Advise the principal on whether the figure of 18 hours should be used in the school-wide report, justifying your answer with reference to what the calculated average actually represents. [2]

Solutions

11QuestionMethods of Data Collection and SamplingAssessment Practice

2 marks~3 minCriterion D

A school wants to estimate overall student satisfaction with cafeteria food. The survey team plans to collect responses only from students present in the cafeteria during lunch.

Explain why this sampling method is biased and how it could lead to a misrepresentation of the entire student population's satisfaction levels. [2]

Solutions

12QuestionReal-Life Applications Games Genetics and Decision-MakingAssessment Practice

4 marks~6 minCriterion C

A genetics laboratory crosses two pea plants and records the phenotypes of 80 offspring.

Tall with green seeds: 30
Tall with yellow seeds: 20
Short with green seeds: 10
Short with yellow seeds: 20

a

Calculate the conditional probability that a randomly selected offspring is tall, given that it has green seeds. [2]

b

A second laboratory claims that height and seed colour are independent traits in this cross. Justify whether the data support this claim. [2]

Solutions

13QuestionConditional Probability IntroductoryAssessment Practice

8 marks~12 minCriterion B

A quality-control technician samples marbles from a production batch. A bag contains red and blue marbles totalling 10. Four compositions are tested by drawing two marbles without replacement.

Initial composition

(R, B)

:
Trial 1:

(4, 6)

— Trial 2:

(5, 5)

— Trial 3:

(6, 4)

— Trial 4:

(7, 3)

a

Calculate

P(\text{second blue} \mid \text{first red})

for each trial. [2]

b

Deduce the general formula for

P(\text{second blue} \mid \text{first red})

in terms of

R

and

B

, showing your reasoning. [2]

c

A new batch has 10 red and 2 blue marbles. The technician's acceptance criterion requires

P(\text{second blue} \mid \text{first red}) > 0.20

. Calculate the probability and advise the technician whether this batch should be accepted or rejected, justifying your answer in context. [4]

Solutions

14QuestionUsing Probability Trees for Multi-Stage EventsAssessment Practice

4 marks~6 minCriterion A

A quality-control engineer randomly selects one of two biased coins from a box. Coin A has

P(\text{H}) = 0.6

and Coin B has

P(\text{H}) = 0.3

, with

P(\text{Coin A}) = 0.4

and

P(\text{Coin B}) = 0.6

. The selected coin is flipped twice. A partially completed probability tree shows the coin selection and first-flip outcomes.

a

Construct the second-flip branches on the tree diagram. [1]

b

Calculate

P(\text{H on first flip})

and

P(\text{HH})

. [2]

c

The sensor flags a fault only when the second flip is Heads given the first was also Heads. Find

P(\text{H on second flip} \mid \text{H on first flip})

and advise the engineer whether this fault signal is reliable enough to act on without further testing. [1]

Solutions

15QuestionUsing 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 equal probability.

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 structure of a probability tree. [2]

c

The engineer runs 4 checks on each chip. Construct a probability tree for this 4-stage process and use your formula to verify the total number of distinct paths. A chip is accepted only if it passes all 4 checks. Advise the engineer whether this acceptance criterion is suitable for a production line, justifying your answer using probability. [3]

Solutions

16QuestionCalculating Mean Median and ModeAssessment Practice

6 marks~9 minCriterion B

A city's air-quality monitoring station records the daily Air Quality Index (AQI). The mean AQI for the first 5 days of a month is 12, and the AQI on day 6 is 18.

a

Calculate

M_6

, the mean AQI for the first 6 days. [2]

b

Deduce a general formula for

M_{n+1}

, the mean of the first

(n+1)

days, in terms of

M_n

,

n

, and

a_{n+1}

. Show all working. [2]

c

The AQI values for days 7 through 10 are 20, 15, 11, and 9. Health guidelines state that a monthly mean AQI above 13 triggers a public advisory. Using your formula from part (b), advise the monitoring station whether a public advisory should be issued after day 10. Justify your answer. [2]

Solutions

17QuestionChoosing the Appropriate MeasureAssessment Practice

4 marks~6 minCriterion D

A school librarian records the number of books read last month by 15 students.

Number of books $(x)$	1	2	3	4	5	20
Frequency $(f)$	4	3	3	2	2	1

a

Calculate the mean number of books read. [1]

b

Determine the median number of books read. [1]

c

The librarian uses the mean to report the "typical" reading behaviour of the group. Advise the librarian whether the mean or the median is the more appropriate measure to use, justifying your answer with reference to the data. [2]

Solutions

18QuestionReal-Life Interpretation of AveragesAssessment Practice

4 marks~6 minCriterion C

A small company employs five people with monthly salaries (in euros) of 2 500, 3 000, 3 200, 3 500, and 4 000. After a performance review, every employee receives a bonus of 500 euros added to their salary.

a

Calculate the mean and median of the original salaries. [2]

b

Calculate the mean and median of the salaries after the bonus is added. [1]

c

The company director claims the bonus "makes salaries fairer by reducing the gap between the lowest- and highest-paid employees." Justify whether the mathematical evidence supports this claim. [1]

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

20QuestionCalculating the interquartile rangeAssessment Practice

4 marks~6 minCriterion C

A school tracks student performance on a standardised mathematics assessment. The box plot shows the distribution of scores for a class of 30 students. The minimum score is 4, the maximum is 40,

Q_1 = 12

, the median

= 18

, and

Q_3 = 28

.

a

Calculate the interquartile range (IQR). [1]

b

Analyse the symmetry of the distribution by calculating the distance from

Q_1

to the median and from the median to

Q_3

. Hence deduce the direction of skew. [2]

c

The school's intervention policy states that the middle 50% of scores must span no more than 20 marks for the group to be considered "consistent." Justify whether this class meets the consistency criterion, and explain what the skewness suggests about the spread of scores in the lower half of the distribution. [1]

Solutions

21QuestionBox and Whisker Plots for Spread of DataAssessment Practice

6 marks~9 minCriterion D

A school uses exam scores from 120 students to decide whether to adjust its teaching programme. The box-and-whisker plot displays the distribution of scores with the following five-number summary:

Minimum

= 28

,

Q_1 = 55

, Median

= 72

,

Q_3 = 88

, Maximum

= 100

a

Calculate the interquartile range (IQR) of the exam scores. [2]

b

Deduce the number of students whose scores fall within the interquartile range and the number whose scores fall within the upper whisker (from

Q_3

to the maximum). [2]

c

The school's policy states that a teaching programme requires adjustment if more than 30% of students score below

Q_1

. Advise the school board whether the teaching programme requires adjustment, justifying your answer with reference to the data. [2]

Solutions

22QuestionConstructing 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

23QuestionCumulative Frequency Graphs and CurvesAssessment Practice

4 marks~6 minCriterion C

A school uses a standardised test (scored 0–100) to assess 200 students for a literacy programme. The cumulative frequency graph shows the distribution of scores.

a

Determine the median and quartiles (

Q_1

and

Q_3

) of the scores. [2]

b

Calculate the interquartile range (IQR). [1]

c

The programme coordinator claims: "More than half the students are eligible for additional support, since most scores fall below 78." Justify whether this claim is mathematically valid, using your results from parts (a) and (b). [1]

Solutions

24QuestionConstructing and Interpreting Box PlotsAssessment Practice

4 marks~6 minCriterion A

A sports analyst is comparing the consistency of two athletes' training scores. She notes that larger sample sizes may produce more stable quartiles. She examines two datasets of daily training scores.

Dataset A (8 values): 42, 45, 48, 50, 52, 55, 58, 60

Dataset B (20 values): 30, 35, 40, 45, 48, 49, 50, 50, 50, 50, 50, 51, 52, 53, 55, 58, 62, 68, 72, 80

a

Calculate the five-number summary and IQR for each dataset. [2]

b

Construct box plots for both datasets on the same scale from 30 to 80. [1]

c

The analyst concludes that Dataset B's scores are more consistent because its IQR is smaller. Critique her conclusion, referring to both the IQR and the range of each dataset. [1]

Solutions

25QuestionCumulative Frequency Graphs and CurvesAssessment Practice

2 marks~3 minCriterion D

A factory produces steel rods with a target length of 200 mm. A cumulative frequency graph is drawn from a sample of 500 rods. The graph passes through the points (198, 50) and (202, 430).

a

Deduce the number of rods outside the acceptable tolerance range of 198–202 mm. [1]

b

Advise the factory manager whether the production process requires corrective action, justifying your answer using the result from part (a). [1]

Solutions

26QuestionCalculating 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

27QuestionSolving Problems using tree diagrams and Venn DiagramsAssessment Practice

2 marks~3 minCriterion C

A school class has 33 students. Let

A

be the set of students who play football and

B

the set of students who play basketball, where

|A| = 15

,

|B| = 10

,

|A \cap B| = 5

, and

|A' \cap B'| = 3

.

Using correct set notation, write one complete sentence that describes the relationship between

A \cup B

and

A' \cap B'

, referring to the universal set

U

of all students in the class. [2]

Solutions

28QuestionCalculating Combined ProbabilitiesAssessment Practice

4 marks~6 minCriterion D

A quality-control technician at a marble factory tests batches by drawing two marbles without replacement from a bag containing 4 red and 2 blue marbles. After 50 trials, the recorded frequencies are:

Outcome	RR	RB	BR	BB
Frequency	18	16	12	4

a

Calculate the experimental probability of each outcome. [1]

b

Construct a tree diagram showing all possible outcomes with their theoretical probabilities, and calculate the theoretical probability of each outcome. [1]

c

Calculate the theoretical and experimental probabilities of drawing "at least one red" marble, and analyse how closely the experimental results match theory. [1]

d

The factory accepts a batch only if the probability of drawing "no reds" is below 0.10. Using your theoretical probabilities, advise the quality-control technician whether this batch should be accepted, and justify your recommendation using the complementary probability. [1]

Solutions

Statistics and Probability — Free MYP5 Mathematics (Standard) Practice Questions