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

What this topic covers

This is the topic examiners lean on hardest, because it's the one place they can test whether you actually understand what a number means, not just whether you can press the right calculator button. A GDC will hand you $r$ , a $p$ -value, or a confidence interval in seconds — the marks live in knowing which tool to reach for, reading the question for the tail direction, and finishing with a sentence back in context. Four threads run through it: describing data you already have, inferring about a population you don't, modelling chance for single events, and modelling chance across a whole range of outcomes. They constantly borrow from each other, so treat this as one connected chapter, not four separate ones.

Read the full Statistics and Probability lesson notes →

1FoundationSAQ-SBinomial distribution and its properties5 marksPaper 1~8 min

A factory produces electronic components. It is known that

8\%

of the components are defective. A quality control inspector selects a random sample of

12

components. The number of defective components,

X

, follows a binomial distribution.

(a)

State one condition, other than a fixed number of trials, that must be satisfied for

X

to follow a binomial distribution. [1 mark]

(b)

Find

P(X = 2)

. [2 marks]

(c)

Given that least one component in the sample is defective, find the probability that exactly two are defective. [2 marks]

Solutions

2FoundationSAQ-SBinomial distribution and its properties7 marksPaper 1~11 min

A fair six-sided die is rolled

8

times. Let

X

be the number of times a score of

4

or greater is obtained.

(a)

State the distribution of

X

, including the values of any parameters. [2 marks]

(b)

Find

P(X = 3)

. [2 marks]

(c)

Given that

X \leq 4

, find the probability that

X = 3

. [3 marks]

Solutions

3MasterySAQ-SBinomial distribution and its properties5 marksPaper 1~8 min

A factory produces electronic components. It is known that

5\%

of components are defective. A quality control inspector selects a random sample of

12

components. Let

X

represent the number of defective components in the sample.

(a)

State two conditions required for

X

to be modelled by a binomial distribution, and identify the parameters

n

and

p

. [2 marks]

(b)

Find the probability that exactly

2

components in the sample are defective. Give your answer correct to 3 significant figures. [3 marks]

Solutions

4ChallengeSAQ-LBinomial distribution and its properties8 marksPaper 1~12 min

A quality control engineer at a pharmaceutical company monitors the production of vaccine vials. Historical data shows that

3\%

of vials have a contamination defect. The company uses a two-stage screening process. - First stage: an automated optical inspection detects

90\%

of contaminated vials but incorrectly flags

2\%

of clean vials as contaminated. Flagged vials are rejected; unflagged vials pass to the second stage. - Second stage: a chemical test, performed only on vials that passed the first stage, correctly identifies

98\%

of contaminated vials and correctly clears

99\%

of clean vials. Vials identified as contaminated in this stage are rejected; all others are released.

(a)

Determine the probability that a randomly selected vial passes the first stage. [2 marks]

(b)

Given that a vial passes the first stage, determine the probability that is contaminated. [3 marks]

(c)

A vial has passed the first stage. Determine the probability that is contaminated and is correctly identified as contaminated by the chemical test in the second stage. > [3 marks]

Solutions

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