Independent and Dependent Events
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Independent and Dependent Events — Free MYP4 Mathematics (Standard) Practice Questions

1QuestionUsing 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 0.20.2, 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(D)=0.2P(\text{D}) = 0.2 at the first stage, and use the multiplication rule to calculate P(ND)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]
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Solutions

2QuestionReal-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(Yellow∣Smooth)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 4060=23\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]
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3QuestionUsing 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 12\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 nn checks, each with mm 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]
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Solutions

4QuestionRecognizing 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 nn marbles in total. Two marbles are drawn in succession without replacement.

nn: 3, 4, 5, 6

P(1st blue)P(\text{1st blue}): 33\dfrac{3}{3}, 34\dfrac{3}{4}, 35\dfrac{3}{5}, 36\dfrac{3}{6}

P(2nd blue∣1st blue)P(\text{2nd blue} \mid \text{1st blue}): 22\dfrac{2}{2}, 23\dfrac{2}{3}, 24\dfrac{2}{4}, 25\dfrac{2}{5}

P(both blue)P(\text{both blue}): 11, 12\dfrac{1}{2}, 310\dfrac{3}{10}, 15\dfrac{1}{5}
a
Deduce a general formula for P(both blue)P(\text{both blue}) in terms of nn. [2]
b
Calculate P(both blue)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 120\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

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