Probability Rules and Tree Diagrams
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Probability Rules and Tree Diagrams — Free MYP5 Mathematics (Standard) Practice Questions

1QuestionSolving Problems using tree diagrams and Venn DiagramsConcept Practice
6 marks~9 minCriterion A
A school survey of 100 students recorded participation in soccer (event AA) and basketball (event BB). 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∩B)n(A \cap B). [1]
b
Calculate n(A)n(A) and n(B)n(B). [2]
c
Using the formula n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B), calculate n(A∪B)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]
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Solutions

2QuestionSolving Problems using tree diagrams and Venn DiagramsAssessment Practice
2 marks~3 minCriterion C
A school class has 33 students. Let AA be the set of students who play football and BB the set of students who play basketball, where ∣A∣=15|A| = 15, ∣B∣=10|B| = 10, ∣A∩B∣=5|A \cap B| = 5, and ∣A′∩B′∣=3|A' \cap B'| = 3.

Using correct set notation, write one complete sentence that describes the relationship between A∪BA \cup B and A′∩B′A' \cap B', referring to the universal set UU of all students in the class. [2]
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Solutions

3QuestionCalculating 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:

OutcomeRRRBBRBB
Frequency1816124
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]
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Solutions

4QuestionCalculating 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 nn red and nn blue marbles, where nn 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 12\dfrac{1}{2}. Advise the factory whether this standard is a useful quality-control threshold, justifying your answer algebraically. [3]

Solutions

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