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Probability Distributions · Paper 1 — Free Maths AI HL Practice Questions

1FoundationSAQ-SBinomial distribution and its properties7 marksPaper 1~11 min
A factory produces electronic components. The probability that a randomly selected component is defective is 0.050.05. A quality control inspector randomly selects 1212 components.
(a)
State two conditions required for the number of defective components, XX, to be modelled by the distribution X∼B(12, 0.05)X \sim B(12,\, 0.05). [2 marks]
(b)
Find P(X=2)P(X = 2). [2 marks]
(c)
Given that least one component in the sample is defective, find the probability that exactly two are defective. [3 marks]
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Solutions

2MasterySAQ-SBinomial distribution and its properties9 marksPaper 1~14 min
A factory produces electronic components. The probability that a randomly selected component is defective is 0.040.04. A quality control inspector selects a random sample of 2020 components and counts the number of defective components, denoted by XX.
(a)
State the distribution of XX, giving two conditions that must be satisfied for this distribution to apply. [2 marks]
(b)
Show that P(X=2)=0.146P(X = 2) = 0.146, correct to three significant figures. [2 marks]
(c)
Find P(X≥2)P(X \geq 2). [2 marks]
(d)
The factory considers a batch of 2020 components to be "acceptable" if fewer than 22 defective components are found. A second factory claims its process reduces the defect probability to 0.020.02. Determine whether the second factory's process significantly increases the probability that a batch is acceptable. Justify your answer. [3 marks]
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Solutions

3ChallengeSAQ-LBinomial distribution and its properties7 marksPaper 1~11 min
A pharmaceutical company is testing a new vaccine. Clinical trials show that the vaccine provides immunity in 85%85\% of individuals who receive it. The company administers the vaccine to a random sample of 20 volunteers from a large population. Let XX be the number of volunteers who develop immunity.
(a)
State the distribution of XX, including its parameters. [1 mark]
(b)
Calculate the probability that exactly 17 of the 20 volunteers develop immunity. [2 marks]
(c)
The company considers the trial a failure if fewer than 16 volunteers develop immunity. Calculate the probability that the trial is declared a failure. [2 marks]
(d)
State one assumption of the binomial distribution required for the model X∼B(20, 0.85)X \sim B(20,\, 0.85) to be valid in this context. [1] (e) Explain how a violation of this assumption could affect the probabilities calculated in parts (b) and (c). [2 marks]
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Solutions

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