Probability Rules and Tree Diagrams
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Probability Rules and Tree Diagrams — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionSolving 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(Red)=410P(\text{Red}) = \dfrac{4}{10}, P(Blue)=610P(\text{Blue}) = \dfrac{6}{10}

Second draw (after Red): P(Red∣Red)=39P(\text{Red} \mid \text{Red}) = \dfrac{3}{9}, P(Blue∣Red)=69P(\text{Blue} \mid \text{Red}) = \dfrac{6}{9}

Second draw (after Blue): P(Red∣Blue)=49P(\text{Red} \mid \text{Blue}) = \dfrac{4}{9}, P(Blue∣Blue)=59P(\text{Blue} \mid \text{Blue}) = \dfrac{5}{9}
a
State P(Red∣Blue)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]
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Solutions

2QuestionTree 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 12\dfrac{1}{2}. Calculate the probability of selecting at least one defective component and justify whether this batch passes or fails inspection. [2]
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Solutions

3QuestionComplementary 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 DD = has the disease and T+T^+ = tests positive.
a
Using the complementary rule and the law of total probability, calculate P(T+)P(T^+), the probability that a randomly selected person tests positive. [2]
b
Calculate P(D∣T+)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]
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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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