Topic 4.09 · SL and HL

Binomial distribution: notes and practice questions

Q: Is Binomial distribution SL or HL?

Both. SL and HL students study Binomial distribution to the same depth.

Q: How do I revise Binomial distribution for IB Maths AI?

Start from the core idea: a discrete random variable's value depends on a random event, unknown until carried out. In the exam: almost always a GDC question: identify n and p, decide between "exactly", "at most" and "at least", then read off the value. The "at least" case needs 1 - P(X ≤ k-1) and the off-by-one is the classic error. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

Q: How does FourtyFive help me practise Binomial distribution?

FourtyFive has 37 Binomial distribution questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Q: Is FourtyFive free for Binomial distribution practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Q: Can I handwrite Binomial distribution answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

15 min read Jump to 20 practice questions

Summary

A discrete random variable's value depends on a random event, unknown until carried out.
A discrete probability distribution lists all possible discrete outcomes and their probabilities.
The Binomial Distribution counts "successes" in a fixed number of independent trials.
Notation: $X \sim B(n, p)$ .
Valid values for $X$ are integers: $0, 1, 2, \dots, n$ .
Conditions for a Binomial Model:
A defined "Trial" (repeated action).
A defined "Success" (desired outcome).
Fixed number of trials ( $n$ ).
Fixed probability of success ( $p$ ) for each trial.
Independent trials.
Binomial Probability Formula (exact $r$ successes):

$P(X=r) = {}^nC_r \times p^r (1-p)^{n-r}$ for $r = 0, 1, \dots, n$

Combinations formula: ${}^nC_r = \frac{n!}{r!(n-r)!}$
Expected Value (Mean): $E(X) = np$
Variance: $\text{Var}(X) = np(1-p)$
To set up a binomial model: Identify $n$ , $p$ , define $X$ in context, state $X \sim B(n, p)$ .
Use GDC's Binomial Cumulative Distribution (Cdf) for probabilities.
GDC input for Cdf: Lower value, Upper value, $n$ , $p$ .
For $P(X \le b)$ : GDC Lower= $0$ , Upper= $b$ .
For $P(X \ge a)$ : GDC Lower= $a$ , Upper= $n$ .
For $P(X=r)$ : Use GDC's Binomial Probability Distribution (Pdf) or Cdf with Lower= $r$ , Upper= $r$ .
Adjust strict inequalities for GDC: $P(a < X < b)$ becomes $P(a+1 \le X \le b-1)$ .
State contextual assumptions for the binomial model (e.g., independence, constant $p$ ).

How it is examined

Almost always a GDC question: identify $n$ and $p$ , decide between "exactly", "at most" and "at least", then read off the value. The "at least" case needs $1 - P(X \le k-1)$ and the off-by-one is the classic error. Justifying that a binomial model is appropriate (fixed trials, two outcomes, constant probability, independence) is a separate mark that students skip.

Given in the booklet

The binomial distribution with its mean $np$ and variance $np(1-p)$ .

Key ideas

The binomial distribution.
The mean and variance of the binomial distribution.

Not assessed

Not required: formal proof of mean and variance.

Linking questions

Aim 8: Pascal's triangle, and attributing the origin of a mathematical discovery to the wrong mathematician.
International-mindedness: the so-called "Pascal's triangle" was known to the Chinese mathematician Yang Hui much earlier than Pascal.
TOK: what criteria can we use to decide between different models?
Enrichment only, so not examinable at SL: hypothesis testing using the binomial distribution. It becomes examinable at HL under AHL 4.18.

Practice questions

37 questions · 28 medium · 9 hard

Showing 20 of 20

Question 1

MediumPaper 1 · calculator8 marks

(a)

A renowned artisanal bakery claims that only 5% of its specialty sourdough loaves have minor cosmetic imperfections (e.g., slight cracks, uneven browning). A local restaurant owner, who regularly purchases these loaves, decides to test this claim. For their latest delivery, the owner inspects a batch of 150 loaves and finds 12 loaves with cosmetic imperfections.

(a) Identify the type of sampling used by the restaurant owner.

[1]

(b)

(b) State the null and alternative hypotheses for this test.

[2]

(c)

[3]

(d)

(d) The restaurant owner performs the test at the 5% significance level. State the conclusion of the test, giving a reason.

[2]

Question 2

HardPaper 1 · calculator6 marks

(a)

(a) A popular cafe uses a specialty espresso machine that has an 85% probability of functioning correctly on any given morning. The probability of the machine functioning correctly on a given morning is independent of it functioning on any other morning.

Find the probability that the espresso machine functions correctly on exactly four mornings in a particular 5-day workweek.

[2]

(b)

(b) If the specialty espresso machine does not function correctly, the cafe owner must decide whether to use a backup machine or close early for the day. The cafe owner closes early if the specialty machine does not function correctly AND the backup machine is not available. There is a 70% probability that the backup machine is available on any given morning, independent of the specialty machine's status or any other morning.

Find the probability that the cafe owner does not have to close early in a particular 5-day workweek.

[4]

Question 3

MediumPaper 1 · calculator7 marks

(a)

A manufacturing plant produces a specific electronic component. Due to the complexity of the manufacturing process, there is a 15% chance that any given component will be defective. A batch of 12 components is randomly selected for quality control inspection. Each component's defect status is independent of others.

(a) Calculate the expected number of components in the batch that are non-defective.

[2]

(b)

(b) Calculate the probability that exactly 3 components in the batch are defective.

[2]

(c)

[3]

Question 4

HardPaper 1 · calculator12 marks

(a)

The lifespan of a certain brand of smartphone battery is normally distributed with a mean of $\mu = 800$ days and a standard deviation of $\sigma = 50$ days.

(a) The probability that a randomly selected battery lasts less than $k$ days is $0.15$ . Find the value of $k$ .

[3]

(b)

(b) A battery is randomly selected. It is known that the battery lasts longer than $750$ days. What is the probability that it lasts longer than $820$ days?

[4]

(c)

(c) A retailer orders $200$ batteries. What is the probability that at most $62$ of them have a lifespan less than $770$ days?

[5]

Question 5

MediumPaper 1 · calculator6 marks

(a)

Leo is a student who takes a short mathematics quiz every day for five days each week. He has a 70% probability of passing any given quiz. The outcome of one quiz is independent of any other quiz.

(a) Find the probability that Leo passes exactly three quizzes in a particular five-day week.

[2]

(b)

Leo studies for his quiz with a 60% probability on any given morning, independent of whether he passes the quiz or not. Leo is considered 'well-prepared' for a quiz if he passes the quiz AND he studied for it.

(b) Find the probability that Leo is not well-prepared on any day in a particular five-day week.

[4]

Question 6

HardPaper 2 · calculator16 marks

(a)

A factory produces specialized electronic components. The total "quality score" of a component, $T$ , is a combination of scores from three independent inspection stages:

Stage 1: Automated visual inspection. The score $X$ from this stage has an expectation of $2.5$ and a standard deviation of $0.7$ .

Stage 2: Manual functional test. A batch of $8$ critical functions are tested, and the score $Y$ is the number of functions that pass. Each function has a $0.35$ probability of passing, independently.

Stage 3: Environmental stress test. The score $Z$ from this stage, representing the number of successful stress cycles, follows a Poisson distribution with a mean of $4.2$ .

The overall quality score for a component is given by $T = X + 2Y + Z$ .

Calculate the expected value and variance of the total quality score $T$ .

[6]

(b)

Given that the distribution of $T$ can be approximated by a Normal distribution, find the probability that a randomly selected component has a total quality score between $10$ and $14$ (inclusive of $10$ , exclusive of $14$ ).

[4]

(c)

The factory manager wants to ensure that the mean total quality score of a sample of $n$ components is within $0.8$ units of the true mean, with a probability of at least $0.95$ . Find the minimum sample size $n$ required.

[6]

Question 7

MediumPaper 1 · calculator8 marks

(a)

A manufacturing company produces electronic components. Historically, the defect rate for a specific component has been 15%. A new production line is implemented, and the quality control manager wants to test if the defect rate has decreased. She assumes that the defect status of each component is independent of others.

(a) Write down suitable hypotheses for this test.

[2]

(b)

(b) The quality control manager decides to take a random sample of 120 components. She will reject the null hypothesis if fewer than 12 components are found to be defective.

Find the probability that she makes a Type I error.

[3]

(c)

Find the probability that she makes a Type II error.

[3]

Question 8

HardPaper 2 · calculator16 marks

(a)

A factory produces electronic components. A quality control inspector randomly selects a batch of $3$ components and checks for defects. This process is repeated for $100$ batches. The number of defective components in each batch is recorded as shown in the table below.

Number of defects ( $x$ )	$0$	$1$	$2$	$3$
Frequency	$50$	$35$	$12$	$3$

The manager claims that the number of defective components in a batch of $3$ follows a binomial distribution $B(3, 0.2)$ .

Show that the expected frequency for a batch having $0$ defective components is $51.2$ .

[3]

(b)

Find the table of expected frequencies for the number of defective components in $100$ batches, assuming the binomial distribution $B(3, 0.2)$ .

[3]

(c)

State the null and alternative hypotheses for a goodness-of-fit test. Justify any necessary adjustments to the categories for the test and state the degrees of freedom.

[4]

(d)

Calculate the Chi-squared test statistic for this data, using the adjusted categories and assuming a $5\%$ significance level. The critical value for this test is $5.991$ .

[4]

(e)

State the conclusion for the test, justifying your answer.

[2]

Question 9

MediumPaper 1 · calculator6 marks

(a)(i)

A company manufactures microchips. On average, 1 in 20 microchips fails the final quality control test. The failure of any microchip is independent of others.

In a randomly selected batch of 30 microchips, calculate the probability that exactly three microchips fail quality control.

[2]

(a)(ii)

In a randomly selected batch of 30 microchips, calculate the probability that more than two microchips fail quality control.

[2]

(b)

The company sells functional microchips for $75 each and can sell failed microchips (for parts/rework) for$ 20 each.

Calculate the expected revenue, in dollars, from selling a randomly selected batch of 30 microchips.

[2]

Question 10

HardPaper 1 · calculator9 marks

(a)

A call centre receives calls at an average rate of $1.8$ calls per minute during a specific period of the day.

The number of calls received in a given minute can be modelled by a Poisson distribution.

(a) Find the probability that the call centre receives exactly one call in a randomly selected minute.

[1]

(b)

(b) Find the probability that the call centre receives at least one call in a randomly selected minute.

[1]

(c)

The call centre supervisor observes the calls received over a period of $4$ independent minutes.

[2]

(d)

(d) Find the probability that exactly one call is received in each of these $4$ minutes.

[2]

(e)

(e) Find the probability that at least one call is received in exactly $2$ of the $4$ minutes.

[3]

Question 11

MediumPaper 1 · calculator7 marks

(a)(i)

9. [Maximum mark: 7]

A factory produces electronic components. Historical data shows that 15% of the components produced are defective.

A quality control inspector randomly selects a batch of 25 components for testing.

As part of its quality control, the factory uses the model $X \sim B(25, 0.15)$ , where $X$ is the number of defective components in a randomly selected batch.

(a) Calculate the

(i) mean of $X$ .

[1]

(a)(ii)

(ii) variance of $X$ .

[1]

(b)(i)

(b) Find the probability that

(i) exactly 3 components are defective.

[2]

(b)(ii)

(ii) fewer than 5 components are defective.

[2]

(c)

[1]

Question 12

HardPaper 2 · calculator14 marks

(a)

The lifespan of a new model of LED light bulb, $L$ , is normally distributed with a mean of $12000$ hours and a standard deviation of $800$ hours.

Sketch a diagram showing this information.

[2]

(b)

Find the proportion of these LED bulbs that have a lifespan between $11000$ hours and $13000$ hours.

[2]

(c)

A large batch of $200$ LED bulbs is purchased. Determine the expected number of bulbs in this batch that will have a lifespan of less than $10800$ hours.

[3]

(d)

It is observed that $15\%$ of the LED bulbs last longer than $h$ hours. Estimate the value of $h$ .

[3]

(e)

Ten of these LED bulbs are chosen at random. Find the probability that exactly two of them last longer than $12800$ hours.

[4]

Question 13

MediumPaper 1 · calculator8 marks

(a)

Alex is taking a multiple-choice quiz with $5$ questions. For each question, there are four possible answers, and Alex guesses randomly. The probability that Alex answers any single question correctly is $0.35$ .

(a) Find the probability that Alex answers at least one question correctly.

[3]

(b)

(b) Find the probability that Alex answers exactly one question correctly.

[2]

(c)

[3]

Question 14

HardPaper 2 · calculator14 marks

(a)(i)

The diameter of a certain type of industrial component is modelled by a normal distribution with a mean of $50$ mm and a standard deviation of $0.5$ mm.

Find the probability that a randomly selected component has a diameter less than $50.7$ mm.

[2]

(a)(ii)

Find the probability that a randomly selected component has a diameter greater than $51.2$ mm.

[1]

(b)

Assuming the diameters of components are independent, find the probability that two consecutive components both have a diameter greater than $51.2$ mm.

[2]

(c)

A component is classified as 'premium' if its diameter is between $49.5$ mm and $50.5$ mm. A batch of three components is considered 'high quality' if all three components in the batch are premium. Find the probability that a randomly selected batch is NOT high quality.

[2]

(d)(i)

In a production run, $12$ batches of three components are produced. Find the probability that at least $7$ of these batches are high quality.

[3]

(d)(ii)

Find the probability that between $7$ and $10$ (exclusive of $10$ ) of these batches are high quality.

[2]

(d)(iii)

Given that at least $7$ batches are high quality, find the probability that less than $10$ batches are high quality.

[2]

Question 15

MediumPaper 1 · calculator11 marks

(a)

A factory produces electronic components. A quality control inspector randomly selects batches of $5$ components and records the number of defective items. Over a month, $120$ such batches were inspected. The observed frequencies are shown in the table below:

Number of defective items	$0$	$1$	$2$ or more
Frequency	$75$	$35$	$10$

It is assumed that the number of defective items in a batch follows a binomial distribution $B(5, 0.1)$ , where $0.1$ is the probability of a single component being defective.

(a) Calculate the expected frequencies for obtaining $0$ , $1$ , and $2$ or more defective items in a batch of $5$ , based on the binomial distribution $B(5, 0.1)$ and a total of $120$ batches.

[4]

(b)

(b) Perform a goodness of fit test at the $5\%$ significance level to determine if the observed data fits the assumed binomial distribution. You should state the null and alternative hypotheses, calculate the $\chi^2$ test statistic, and justify your conclusion. The critical value for this test is $5.991$ .

[7]

Question 16

HardPaper 2 · calculator14 marks

(a)

(a) The number of customer service emails received by 'TechSolutions' per hour follows a Poisson distribution with a mean of $2.8$ . Using this model, find the probability that TechSolutions receives exactly $3$ emails in a given hour.

[1]

(b)(i)

(b) Over a period of $4$ consecutive hours, find the probability that TechSolutions receives:

(i) exactly $10$ emails.

[2]

(b)(ii)

(ii) emails during the first and third hour only (i.e., at least one email in the first hour, no emails in the second hour, and at least one email in the third hour).

[3]

(c)

(c) Over a working day of $10$ hours, find the probability that there are exactly $2$ hours during which TechSolutions receives no emails.

[4]

(d)

(d) TechSolutions expands its operations and opens a new department, 'SupportPlus', which also receives emails. The number of emails received by each 'SupportPlus' agent per hour follows a Poisson distribution with a mean of $1.5$ . Assuming these are independent of the main TechSolutions department and each other, determine the least number of SupportPlus agents required so that the total probability of receiving at least $25$ emails across all departments (main TechSolutions and the SupportPlus agents) in an hour is greater than $0.15$ .

[4]

Question 17

MediumPaper 2 · calculator15 marks

(a)

A factory produces light bulbs, and it is known that the probability of a light bulb being defective is $0.2$ . A quality control inspector takes a sample of $3$ light bulbs from a production line and records the number of defective bulbs. This process is repeated $100$ times, and the observed frequencies are shown in the table below.

Number of defective bulbs	$0$	$1$	$2$	$3$
Frequency	$48$	$40$	$10$	$2$

Show that the expected frequency for getting $0$ defective bulbs in a sample of $3$ is $51.2$ .

[2]

(b)

Find the table of expected frequencies, pooling categories as necessary to ensure all expected frequencies are at least $5$ .

[4]

(c)

State the null and alternative hypotheses for a goodness of fit test.

[2]

(d)

Write down the number of degrees of freedom.

[1]

(e)

Calculate the $\chi^2$ test statistic.

[3]

(f)

The critical value for this test at the $5\%$ significance level is $5.991$ .

State the conclusion for the test and give a reason for your answer.

[3]

Question 18

HardPaper 2 · calculator15 marks

(a)

The 'SoundWave' concert hall has a capacity of $100$ seats. Historical data shows that $90\%$ of ticket holders attend the concert. The management decides to sell $105$ tickets, hoping that no more than $100$ concertgoers will arrive.

The number of concertgoers who arrive is assumed to follow a binomial distribution with a probability of $0.9$ .

(a) Calculate the probability that more than $100$ concertgoers arrive for the performance.

[3]

(b)(i)

(b) (i) Write down the expected number of concertgoers who will arrive if $100$ tickets are sold.

[2]

(b)(ii)

(ii) Find the maximum number of tickets that could be sold if the expected number of concertgoers who arrive must be less than or equal to $100$ .

[2]

(c)

Each ticket costs $120. If more concertgoers arrive than there are seats, the venue will give$ 200 in compensation to each concertgoer who cannot be seated.

(c) Find, to the nearest integer, the expected increase or decrease in the money made by the venue if they decide to sell $105$ tickets rather than $100$ .

[8]

Question 19

MediumPaper 1 · calculator5 marks

(a)

The volume of liquid in bottles produced by a beverage company is normally distributed with a mean of $750$ mL and a standard deviation of $5$ mL. A bottle is considered 'underfilled' if its volume is less than $740$ mL.

(a) Find the percentage of bottles that are classified as underfilled.

[2]

(b)

Bottles are packed into crates, with each crate containing $12$ bottles.

(b) Find the probability that a random crate has at most two underfilled bottles.

[3]

Question 20

MediumPaper 2 · calculator12 marks

(a)

(a) Show that there are $12$ factors of $96$ .

[3]

(b)(i)

(b) A player spins a "Factor Frenzy" spinner $8$ times. The spinner has numbers from $1$ to $96$ . A player scores a "bonus point" if the number spun is a factor of $96$ .

(i) Find the probability that the player scores exactly three bonus points.

[3]

(b)(ii)

(ii) Find the probability that the player scores at least three bonus points.

[2]

(b)(iii)

(iii) Find the probability that the player scores at most three bonus points.

[2]

(b)(iv)

(iv) Find the probability that the player scores bonus points on the first three spins, and does not score a bonus point on the fourth spin.

[2]

17 more Binomial distribution questions in the app

Every answer is marked mark by mark, IB-style, and the AI tutor helps when you are stuck.

Where marks are lost

Using your own wrong value after failing a "show that." All follow through is withdrawn for the rest of that question.

FAQ

Questions,
answered.

Can't find what you're looking for? Email our student team.

What does Binomial distribution cover in IB Maths AI?

A discrete random variable's value depends on a random event, unknown until carried out. A discrete probability distribution lists all possible discrete outcomes and their probabilities. The Binomial Distribution counts "successes" in a fixed number of independent trials.

Is Binomial distribution SL or HL?

Both. SL and HL students study Binomial distribution to the same depth.

How do I revise Binomial distribution for IB Maths AI?

Start from the core idea: a discrete random variable's value depends on a random event, unknown until carried out. In the exam: almost always a GDC question: identify n and p, decide between "exactly", "at most" and "at least", then read off the value. The "at least" case needs 1 - P(X ≤ k-1) and the off-by-one is the classic error. Then practise exam-style questions, easiest first, writing out every step of your working before you check it.

How does FourtyFive help me practise Binomial distribution?

FourtyFive has 37 Binomial distribution questions. Every answer you write is marked mark by mark, IB-style, and you see where each mark was won or lost. Every part has a hint, the AI tutor helps you through the step you are stuck on, and your Study Profile picks what to practise next.

Is FourtyFive free for Binomial distribution practice?

Yes. A free account gives you 50 marked answers a month, and you do not need a card to sign up.

Can I handwrite Binomial distribution answers on an iPad?

Yes. In the FourtyFive iPad app you write your working by hand with Apple Pencil, the way you would on paper, and it is marked the same way.

Binomial distribution: notes and practice questions

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

17 more Binomial distribution questions in the app

Where marks are lost

Practise this topic as a session

Questions,
answered.

Start with the IB question
bank built for you.

How it is examined

Linking questions

Practice questions

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19

Question 20

17 more Binomial distribution questions in the app

Where marks are lost

Questions,answered.

Start with the IB questionbank built for you.

Questions,
answered.

Start with the IB question
bank built for you.