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SL 4.8—Binomial distribution - IB Questionbank | RevisionDojo

Practice SL 4.8—Binomial distribution with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

A factory, producing plastic gifts for a fast food restaurant’s Jolly meals, claims that just $1 %$ ofthe toys produced are faulty.

A restaurant manager wants to test this claim. A box of $200$ toys is delivered to the restaurant.The manager checks all the toys in this box and four toys are found to be faulty.

The restaurant manager performs a one-tailed hypothesis test, at the $10 %$ significance level,to determine whether the factory’s claim is reasonable. It is known that faults in the toysoccur independently.

Identify the type of sampling used by the restaurant manager.

[1]

Write down the null and alternative hypotheses.

[2]

Find the $p$ -value for the test.

[2]

State the conclusion of the test. Give a reason for your answer.

[2]

Question 2

HLPaper 2

The number of marathons that Audrey runs in any given year can be modelled by a Poisson distribution with mean 1.3 .

Calculate the probability that Audrey will run at least two marathons in a particular year.

[2]

Find the probability that she will run at least two marathons in exactly four out of the following five years.

[4]

Question 3

HLPaper 2

A Chocolate Shop advertises free gifts to customers that collect three vouchers. The vouchers are placed at random into 10% of all chocolate bars sold at this shop. Kati buys some of these bars and she opens them one at a time to see if they contain a voucher. Let ${\text{P}}(X = n)$ be the probability that Kati obtains her third voucher on the $n{\text{th}}$ bar opened.

(It is assumed that the probability that a chocolate bar contains a voucher stays at 10% throughout the question.)

It is given that ${\text{P}}(X = n) = \frac{{{n^2} + an + b}}{{2000}} \times {0.9^{n - 3}}$ for $n \geqslant 3,{\text{ }}n \in \mathbb{N}$ .

Kati’s mother goes to the shop and buys $x$ chocolate bars. She takes the bars home for Kati to open.

Show that ${\text{P}}(X = 3) = 0.001$ and ${\text{P}}(X = 4) = 0.0027$ .

[3]

Find the values of the constants $a$ and $b$ .

[5]

Deduce that $\frac{{{\text{P}}(X = n)}}{{{\text{P}}(X = n - 1)}} = \frac{{0.9(n - 1)}}{{n - 3}}$ for $n > 3$ .

[4]

(i) Hence show that $X$ has two modes ${m_1}$ and ${m_2}$ .

(ii) State the values of ${m_1}$ and ${m_2}$ .

[5]

Determine the minimum value of $x$ such that the probability Kati receives at least one free gift is greater than 0.5.

[3]

Question 4

SLPaper 1

Mr Burke teaches a mathematics class with 15 students. In this class there are 6 femalestudents and 9 male students.

Each day Mr Burke randomly chooses one student to answer a homework question.

In the first month, Mr Burke will teach his class 20 times.

Find the probability that on any given day Mr Burke chooses a female student toanswer a question.

[1]

Find the probability he will choose a female student 8 times.

[2]

Find the probability he will choose a male student at most 9 times.

[3]

Question 5

HLPaper 2

The weights, in grams, of individual packets of coffee can be modelled by a normal distribution,with mean $102 g$ and standard deviation $8 g$ .

Find the probability that a randomly selected packet has a weight less than $100 g$ .

[2]

The probability that a randomly selected packet has a weight greater than $w$ gramsis $0.444$ . Find the value of $w$ .

[2]

A packet is randomly selected. Given that the packet has a weight greater than $105 g$ ,find the probability that it has a weight greater than $110 g$ .

[3]

From a random sample of $500$ packets, determine the number of packets that would beexpected to have a weight lying within $1.5$ standard deviations of the mean.

[3]

Packets are delivered to supermarkets in batches of $80$ . Determine the probability thatat least $20$ packets from a randomly selected batch have a weight less than $95 g$ .

[4]

Question 6

HLPaper 2

Jenna is a keen book reader. The number of books she reads during one week can bemodelled by a Poisson distribution with mean $2.6$ .

Determine the expected number of weeks in one year, of $52$ weeks, during which Jenna reads at least four books.

[4]

Question 7

HLPaper 2

When carpet is manufactured, small faults occur at random. The number of faults in Deluxe carpets can be modelled by a Poisson distribution with mean 0.5 faults per 20 $\,$ m². Mr Smith chooses Deluxe carpets to replace the carpets in his office building. The office building has 10 rooms, each with the area of 80 $\,$ m².

Find the probability that the carpet laid in the first room has fewer than three faults.

Find the probability that exactly seven rooms will have fewer than three faults in the carpet.

Question 8

HLPaper 2

A continuous random variable $X$ has probability density function $f$ given by

$f(x) = \left\{ {\begin{array}{*{20}{l}} {\frac{{{x^2}}}{a} + b,}&{0 \leqslant x \leqslant 4} \\ 0&{{\text{otherwise}}} \end{array}} \right.{\text{where }}a{\text{ and }}b{\text{ are positive constants.}}$

It is given that ${\text{P}}(X \geqslant 2) = 0.75$ .

Eight independent observations of $X$ are now taken and the random variable $Y$ is the number of observations such that $X \geqslant 2$ .

Show that $a = 32$ and $b = \frac{1}{{12}}$ .

[5]

Find ${\text{E}}(X)$ .

[2]

Find ${\text{Var}}(X)$ .

[2]

Find the median of $X$ .

[3]

Find ${\text{E}}(Y)$ .

[2]

Find ${\text{P}}(Y \geqslant 3)$ .

[1]

Question 9

HLPaper 2

The random variable X has a binomial distribution with parameters n and p.
It is given that E(X) = 3.5.

Find the least possible value of n.

[2]

It is further given that P(X ≤ 1) = 0.09478 correct to 4 significant figures.

Determine the value of n and the value of p.

[5]

Question 10

HLPaper 1

Mr Burke teaches a mathematics class with 15 students. In this class there are 6 femalestudents and 9 male students.

Each day Mr Burke randomly chooses one student to answer a homework question.

In the first month, Mr Burke will teach his class 20 times.

Find the probability he will choose a female student 8 times.

[2]

The Head of Year, Mrs Smith, decides to select a student at random from the year group toread the notices in assembly. There are 80 students in total in the year group. Mrs Smithcalculates the probability of picking a male student 8 times in the first 20 assemblies is0.153357 correct to 6 decimal places.

Find the number of male students in the year group.

[4]

SL 4.8—Binomial distribution Questionbank