Probability and Outcomes — Free IB MYP2 Mathematics Questions

1QuestionComparing Theoretical and Experimental ResultsConcept Practice

2 marks~3 minCriterion A

A spinner is divided into four sectors labelled A, B, C, and D. Sector A has a central angle of

90°

. Theoretical probability is the mathematically expected chance of an outcome.

a

State the theoretical probability of the spinner landing on sector A. [1]

b

Calculate, showing your working, the theoretical probability of landing on sector A. [1]

Solutions

2QuestionSolving Real-Life Problems with Combined EventsConcept Practice

2 marks~3 minCriterion B

For combined events, the total number of outcomes can be found using a pattern.

- Two dice:

6 \times 6 = 36

outcomes
- One die and one coin:

6 \times 2 = 12

outcomes
- Two coins:

2 \times 2 = 4

outcomes

a

Identify the pattern used to find the total number of outcomes for combined events. [1]

b

Calculate the total number of outcomes when spinning a spinner with 4 equal sections and rolling a die. [1]

Solutions

3QuestionCalculating Combined Probabilities BasicConcept Practice

2 marks~3 minCriterion A

A spinner has 4 equal sections: red, blue, green, and yellow. A fair coin is tossed at the same time.

a

State the probability of landing on red. [1]

b

The product rule states that for two independent events (where one outcome does not affect the other), you multiply their probabilities. Calculate the probability of landing on red and getting heads. [1]

Solutions

4QuestionUsing Spinners to Model ProbabilityConcept Practice

2 marks~3 minCriterion A

A spinner is divided into 6 equal sectors: 2 red, 1 blue, 2 green, and 1 yellow.

a

State the probability of landing on blue as a fraction in its simplest form. [1]

b

Explain why the probability of landing on red is greater than the probability of landing on yellow. [1]

Solutions

5QuestionCalculating Basic Probabilities Using Tree DiagramsConcept Practice

2 marks~3 minCriterion A

A spinner has two sections,

A

and

B

. The probability of landing on

A

is

0.6

and the probability of landing on

B

is

0.4

for each spin. The spinner is spun twice.

a

Identify the probability of landing on

A

on the first spin and

B

on the second spin. [1]

b

Calculate

P(A \text{ then } B)

. [1]

Solutions

6QuestionDescribing Events as Likely or UnlikelyConcept Practice

2 marks~3 minCriterion A

A spinner is divided into 4 equal sectors: red, blue, green, and yellow.

a

State the probability of the spinner landing on red. [1]

b

Identify whether landing on red is likely, unlikely, certain, or impossible. Give one reason for your answer. [1]

Solutions

7QuestionCounting Outcomes for Combined EventsConcept Practice

2 marks~3 minCriterion A

A coin is flipped twice. The tree diagram shows all possible outcomes.

a

State the number of outcomes for each single flip. [1]

b

Using the tree diagram, identify all distinct outcomes for two flips and state the total number of outcomes. [1]

Solutions

8QuestionUnderstanding Variability in ResultsAssessment Practice

8 marks~12 minCriterion C

A TV game show uses a coin flip to decide a prize winner. The host flips a coin 10 times and records 7 heads. The contestant requests a fairer test; 100 flips are made, giving 58 heads.

a

State the theoretical probability of getting heads on a fair coin. [1]

b

Calculate the experimental probability of heads for each set of flips. Show your working. [3]

c

Compare the two sets of results. In your answer, explain which result is more reliable for deciding whether the coin is fair, and why the result after 100 flips still does not match the theoretical probability exactly. [4]

Solutions

9QuestionComparing Theoretical and Experimental ResultsAssessment Practice

7 marks~11 minCriterion B

A fair six-sided die is rolled 200 times.

Trials	10	50	200
Number of 3s rolled	2	9	38

a

Calculate the relative frequency of rolling a 3 after each number of trials. Give each answer as a decimal rounded to three decimal places. [3]

b

The theoretical probability of rolling a 3 is

\frac{1}{6} \approx 0.167

. Calculate the difference between each relative frequency and

0.167

. Describe what happens to this difference as the number of trials increases. [2]

c

Compare the relative frequency you would expect after 500 trials with the theoretical probability of

0.167

. Justify your answer. [2]

Solutions

10QuestionConducting Simple Probability ExperimentsAssessment Practice

8 marks~12 minCriterion D

A toy company claims its dice are fair. For a fair die, the theoretical probability of rolling a 6 is

\dfrac{1}{6} \approx 0.167

.

Quality control tests record how often a 6 appears over several series of rolls.

Results:
Series 1: 12 sixes in 50 rolls
Series 2: 18 sixes in 100 rolls
Series 3: 85 sixes in 500 rolls

a

Calculate the experimental probability of rolling a 6 in Series 1. [1]

b

Calculate the experimental probability of rolling a 6 in Series 2 and Series 3. Show both calculations. [3]

c

Compare the three experimental probabilities with the theoretical probability

\dfrac{1}{6}

. Use your results to explain whether the die appears fair and how the number of rolls affects the reliability of the estimate. [4]

Solutions

11QuestionProbability of a Single Event OccurringAssessment Practice

4 marks~6 minCriterion C

A spinner has three equal sections: Red, Blue, and Green. Each colour has a theoretical probability of

\dfrac{1}{3}

of being landed on. A student spins the spinner 30 times. Red appears 18 times.

a

Calculate the experimental probability of landing on Red. [1]

b

Explain why this result does not prove the spinner is unfair, referring to sample size in your answer. [1]

c

Compare the experimental and theoretical probabilities and describe what would happen to the experimental probability if the spinner were spun 300 times instead. [2]

Solutions

12QuestionIntroduction to Independent EventsAssessment Practice

2 marks~3 minCriterion D

A football coach selects a captain, returns that player's name to the pool, then selects a vice-captain from the same 12 players.

a

Identify one assumption the independent events model makes about this selection process. [1]

b

Explain one reason why this model does not fully reflect real-world team selection. [1]

Solutions

13QuestionProbability of Rolling a DieAssessment Practice

5 marks~8 minCriterion B

A student rolls two fair dice 20 times and records the sum each time.

Sum	2	3	4	5	6	7	8	9	10	11	12
Frequency	1	2	3	3	4	5	4	3	2	2	1

a

State the relative frequency (as a decimal) for each sum. [1]

b

Identify the most frequent and least frequent sums from the table. Then explain, in one sentence, why sums near 7 are more likely than sums of 2 or 12 when rolling two dice. [2]

c

List all possible pairs of dice outcomes that give a sum of 2, a sum of 7, and a sum of 12. Compare the number of pairs for each sum and explain how this supports your answer to part (b). [2]

Solutions

14QuestionCreating Fair and Biased ModelsAssessment Practice

2 marks~3 minCriterion D

A quality control worker rolls a six-sided die 600 times. The number 3 appears 90 times.

a

Calculate the expected number of times 3 should appear if the die is fair. [1]

b

Using your answer to part (a), explain whether 600 rolls is enough to conclude that the die is biased. [1]

Solutions

15QuestionProbability of Flipping a CoinAssessment Practice

4 marks~6 minCriterion C

A student claims that a fair coin will always land heads up exactly 5 times out of 10 flips because the theoretical probability of heads is $\frac{1}{2}$ . The student flips a coin 10 times and gets 7 heads and 3 tails.

a

[2 marks] Discuss whether the student's mathematical model (theoretical probability) is valid, explaining why the actual outcome differs from the expected 5 heads.

b

[2 marks] Explain one limitation of using the theoretical probability model to predict the outcome of a single short experiment like this one.

Solutions

16QuestionExploring Overlapping and Disjoint EventsAssessment Practice

2 marks~3 minCriterion D

A school cafeteria manager records that 60 students buy pizza, 30 buy salad, and 15 buy both.

a

Describe one way this data helps the manager decide how much food to prepare. [1]

b

Identify one limitation of using a Venn diagram to plan food orders, and explain why it creates a problem. [1]

Solutions

17QuestionInterpreting Simple Venn DiagramsAssessment Practice

6 marks~9 minCriterion B

A Venn diagram shows two sets,

A

and

B

, inside a universal set

\xi

of 20 elements, where

n(A) = 10

and

n(B) = 8

.

a

The intersection

A \cap B

(the overlap of

A

and

B

) contains 6 elements. Calculate

n(A \cup B)

, the total number of elements in

A

or

B

. [2]

b

The table below shows four diagrams with different intersection sizes.

n(A \cap B)

: 2, 4, 6, 8

n(A \cup B)^{\prime}

(complement): ?, ?, ?, ?

Calculate the complement size for each intersection value. Describe the pattern you notice. [3]

c

A fifth diagram has

n(A \cap B) = 5

. Using your pattern, estimate the complement size and explain why your answer is reasonable. [1]

Solutions

18QuestionUnderstanding Tree Diagrams for Two EventsAssessment Practice

5 marks~8 minCriterion C

A fair spinner has three equal sections labelled A, B, and C. It is spun twice. A student's tree diagram shows the probability of A on the first spin as

\frac{1}{2}

and on the second spin as

\frac{1}{3}

, giving

P(\text{A and A}) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}

.

a

Identify the error in the tree diagram. [1]

b

Calculate the correct probability of getting A on both spins. Show your working using the multiplication rule for independent events (where the result of one spin does not affect the other). [2]

c

A classmate says: "It doesn't matter which spin has the wrong probability — the final answer

\frac{1}{6}

is close enough to

\frac{1}{9}

, so the error is not important." Compare the two probabilities and explain why the error does matter. [2]

Solutions

19QuestionProbability Scale from 0 to 1Assessment Practice

4 marks~6 minCriterion B

A spinner has four equal sections: red, blue, green, and yellow. The table below shows how many times green appeared in repeated spins.

Number of spins	10	50	100	200
Green appeared	4	14	25	50

a

Calculate the experimental probability of landing on green for each number of spins. [2]

b

As the number of spins increases, the experimental probability gets closer to the theoretical probability. Use this idea to explain what the theoretical probability of landing on green is, and why. [2]

Solutions

20QuestionDescribing Events as Likely or UnlikelyAssessment Practice

6 marks~9 minCriterion D

A weather app predicts an 80% probability of rain tomorrow. It compared tomorrow's forecast conditions to past weather records and found that rain occurred on 80 out of 100 similar days.

a

State what the fraction

\dfrac{80}{100}

represents in this context. [1]

b

Explain how the app used those 100 similar days to calculate the 80% probability. Show your working. [2]

c

A school picnic and an outdoor wedding are both planned for tomorrow. Compare how the two groups should respond to the 80% rain probability, and explain why their decisions may differ. [3]

Solutions

21QuestionDescribing Events as Likely or UnlikelyAssessment Practice

4 marks~6 minCriterion C

A fair spinner has two equal sections: red and blue. A student says, "The spinner has a

50\%

chance of landing on red, so it will land on red exactly 5 times out of 10 spins."

a

Identify one reason why the student's statement is incorrect. [1]

b

Explain why theoretical probability (the mathematically expected result) may not match the actual results in a short series of spins. [3]

Solutions

22QuestionCounting Outcomes for Combined EventsAssessment Practice

2 marks~3 minCriterion B

A 6-sided die is rolled and some coins are flipped. The table below shows the total number of possible outcomes.

Number of coins	1	2	3
Total outcomes	12	24	48

a

Identify the pattern in the total outcomes as the number of coins increases. [1]

b

State the total outcomes for 4 coins and write the general rule for finding the total outcomes when rolling one 6-sided die and flipping

n

coins. [1]

Solutions

23QuestionCounting Outcomes for Combined EventsAssessment Practice

6 marks~9 minCriterion C

A board game uses two fair six-sided dice. A player must roll a total of 7 to start. The rulebook claims: "All totals from 2 to 12 are equally likely."

a

Draw a sample space diagram showing all 36 possible outcomes. Use it to calculate the probability of rolling a total of 7. [2]

b

Explain why the rulebook's claim is incorrect. Use values from your diagram to support your answer. [2]

c

A second set of dice is slightly weighted so each die is more likely to land on 6. Compare how this would affect the probability of rolling a total of 7, and state whether your diagram from part (a) would still be useful. [2]

Solutions

24QuestionCounting Outcomes for Combined EventsAssessment Practice

8 marks~12 minCriterion D

A school canteen offers 3 main dishes, 2 sides, and 4 drinks. The manager says there are exactly 24 possible meal combinations.

a

State the number of choices for each item and calculate the total number of meal combinations. [2]

b

Explain two real-world factors that could reduce the number of practical meal combinations. [2]

c

Compare the usefulness of the 24-combination count for the manager's planning with its limitations as a real-world model. [4]

Solutions

Probability and Outcomes — Free MYP2 Mathematics Practice Questions