4.8 The binomial distribution questions for IB Maths AA

58 questions on The binomial distribution, written by Augment in the style of IB exam papers. Six of them are below in full, to answer on paper.

IB Mathematics: Analysis and Approaches
Part of 4 Statistics and probability
SL and HL

Questions by paper and level

SL and HL questions are for students at either level. HL only questions cover what only HL students study.

Paper	SL and HL	HL only	All
Paper 1	8	1	9
Paper 2	43	5	48
Paper 3HL only	0	1	1
Every paper	51	7	58

Counted on 10 November 2026

Questions to try

Written by Augment in the style of IB exams, not past papers. Answer them on paper, then check each one against its markscheme with a free account.

Question 1 · AA-00596Paper 1 · 6 marks

A fair coin is tossed $n$ times, where $n \geq 2$ . Let $X$ be the number of heads obtained.

(a)
Show that $P (X \leq 1) = \frac{n + 1}{2^{n}}$ .
3 marks
(b)
Find the least value of $n$ such that the probability of obtaining at least two heads is greater than 0.99.
3 marks

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Question 2 · AA-00390Paper 2 · 14 marks

A town has 6400 households. An energy company selects a simple random sample of 250 of these households. Of the households in the sample, 48 have solar panels.

(a)
Estimate the number of households in the town that have solar panels.
2 marks
(b)(i)
The daily electricity use, $E$ kWh, of each household in the sample is recorded. The results are shown in the following table.
Daily use, $E$ kWh
Number of households
$0 \leq E < 5$
18
$5 \leq E < 10$
52
$10 \leq E < 15$
85
$15 \leq E < 20$
61
$20 \leq E < 25$
26
$25 \leq E < 30$
8
Write down the modal class.
1 mark
(b)(ii)
Find an estimate of the mean and an estimate of the standard deviation of the daily electricity use.
3 marks
(c)
Estimate the number of households in the town that use at least 20 kWh per day.
2 marks
(d)
The company contacts 12 households in the town, chosen at random. Assume that the proportion of all households in the town with solar panels is the same as the proportion in the sample. Find the probability that at least 3 of the 12 households have solar panels.
3 marks
(e)
State one assumption required for the model used in part (d).
1 mark
(f)
An employee suggests that it would have been quicker to survey the 250 households that live closest to the company's office. Give two reasons why this would be likely to give less reliable estimates than the random sample.
2 marks

Daily use, $E$ kWh	Number of households
$0 \leq E < 5$	18
$5 \leq E < 10$	52
$10 \leq E < 15$	85
$15 \leq E < 20$	61
$20 \leq E < 25$	26
$25 \leq E < 30$	8

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Question 3 · AA-00460Paper 3 · HL only · 28 marks

This question asks you to investigate how the probability that a component is faulty changes as the results of repeated tests are obtained.

A factory makes electronic components. Each component is either faulty ( $F$ ) or good ( $F^{'}$ ). A test on a component gives one of two results, "fail" or "pass". The test has accuracy $a$ , where $\frac{1}{2} < a < 1$ : a faulty component fails the test with probability $a$ , and a good component passes the test with probability $a$ .

When a component is tested more than once, the results are independent of each other, given whether or not the component is faulty.

For any event $E$ with $0 < P (E) < 1$ , the odds of $E$ are defined as $O (E) = \frac{P (E)}{P (E^{'})}$ . Similarly, $O (E ∣ R) = \frac{P (E ∣ R)}{P (E^{'} ∣ R)}$ for an event $R$ .

(a)
Consider the case $P (F) = 0.1$ and $a = 0.8$ . A component is tested once and fails. Find the probability that it is faulty.
3 marks
(b)
Show that $O (F ∣ fail) = \frac{a}{1 - a} \times O (F)$ .
3 marks
(c)
Find a similar expression for $O (F ∣ pass)$ in terms of $a$ and $O (F)$ .
2 marks
(d)(i)
Write down $P (R_{k} ∣ F)$ and $P (R_{k} ∣ F^{'})$ .
2 marks
(d)(ii)
Hence show that $O (F ∣ R_{k}) = {(\frac{a}{1 - a})}^{k} \times O (F)$ .
3 marks
(e)
A component is tested $m + n$ times. It fails $m$ of the tests and passes the other $n$ , in some order. Let $S$ be the event that these results are obtained, in that order. Show that $O (F ∣ S) = {(\frac{a}{1 - a})}^{m - n} \times O (F)$ .
3 marks
(f)
Explain briefly the significance of the result in part (e) for a component that fails exactly as many tests as it passes.
2 marks
(g)
A component is tested 10 times. It fails 6 tests and passes 4. Find the probability that it is faulty.
3 marks
(h)
A component is tested repeatedly and fails every test. Find the least number of tests after which the probability that it is faulty is greater than 0.999.
3 marks
(i)
A faulty component is tested 10 times. Find the probability that, from its results, the probability that it is faulty is calculated to be greater than 0.5.
4 marks

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Question 4 · AA-00597Paper 1 · 5 marks

The random variable $X$ follows a binomial distribution with $X \sim B (n, p)$ . It is given that $E (X) = 12$ and $V a r (X) = 7.5$ .

(a)
Find the value of $p$ and the value of $n$ .
4 marks
(b)
Write down an expression for $P (X = 0)$ .
1 mark

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Question 5 · AA-00415Paper 2 · 7 marks

Each time Kemal takes a free throw in basketball, the probability that he scores is 0.72, independently of all other free throws.

In a training session, Kemal takes 12 free throws. Let $X$ be the number of free throws he scores.

(a)
Find $P (X \geq 8)$ .
2 marks
(b)
Given that Kemal scores at least 8 free throws, find the probability that he scores exactly 10.
3 marks
(c)
Kemal scores with each of his first two free throws. Find the probability that he scores at least 8 of the 12 free throws.
2 marks

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Question 6 · AA-00598Paper 1 · 5 marks

A spinner lands on red with probability $\frac{2}{5}$ . The spinner is spun three times and the results of the spins are independent. Let $X$ be the number of times the spinner lands on red.

(a)
Find $P (X = 0)$ .
1 mark
(b)
Find $P (X = 2)$ .
2 marks
(c)
Find $E (X)$ and $V a r (X)$ .
2 marks

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