SL 4.7—Discrete random variables Questionbank

Practice SL 4.7—Discrete random variables with authentic IB Mathematics Analysis and Approaches (AA) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like functions and equations, calculus, complex numbers, sequences and series, and probability and statistics. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

SLPaper 2

A discrete random variable $W$ models the number of customers arriving at a counter in a 10 -minute period, with the probability distribution:

$w$	0	1	4
$P(W=w)$	0.2	$p$	$0.3 p$

(a) find the value of $p$ and the probability distribution. (b) find $E(W)$ and sketch the probability distribution.

find the value of $p$ and the probability distribution.

[3]

find $E(W)$ and sketch the probability distribution.

[3]

Question 2

SLPaper 1

A game involves rolling a fair six-sided die until a 6 appears. Let $X$ be the discrete random variable representing the number of rolls needed to get the first 6 . The probability distribution is given by $P(X=n)=\frac{5^{n-1}}{6^{n}}$ for $n=1,2,3, \ldots$ .

Verify that $P(X=1)=\frac{1}{6}$ and $P(X=2)=\frac{5}{36}$ .

[2]

Show that the expected number of rolls, $E(X)$ , is 6 .

[3]

Find the probability that at least 4 rolls are needed to get the first 6 .

[3]

Question 3

SLPaper 1

A discrete random variable $X$ represents the number of successful attempts in a game, with the probability distribution given by the following table.

$x$	0	1	2	3
$P(X=x)$	$\frac{1}{5}$	$p$	$\frac{1}{10}$	$q$

Given that the expected value $E(X)=\frac{7}{5}$ ,

find the values of $p$ and $q$ .

[4]

calculate the variance $\operatorname{Var}(X)$ .

[3]

Question 4

SLPaper 1

A charity raffle sells tickets, where each ticket has a 5

Show that $P(W=2)=0.0025$ and find $P(W=3)$ .

[2]

Find $E(W)$ and the probability that at most 10 tickets are needed.

[4]

If the prize is not claimed, its value in week $k$ is $100 \cdot 2^{k-1}$ dollars. Find the smallest $k$ such that the expected prize value for a ticket purchased in week $k$ exceeds 10 dollars.

[5]

Prove that the probability distribution is unimodal and find the mode.

[4]

Question 5

SLPaper 1

A discrete random variable $Z$ represents the number of defective items in a sample of 5 , where each item has a 10

$z$	0	1	2	3	4
$P(Z=z)$	$k$	0.2	0.1	0.008	$a$

Find the values of $k$ and $a$ .

[3]

Calculate $E(Z)$ and $\operatorname{Var}(Z)$ .

[4]

Find the probability that at least 2 defective items are found.

[2]

Question 6

SLPaper 1

A discrete random variable $V$ represents the number of correct answers in a 4 -question quiz, where each question has a 25

Write the probability distribution table for $V$ .

[3]

Calculate $E(V)$ and $\operatorname{Var}(V)$ .

[2]

Find the probability that at least 3 questions are answered correctly.

[2]

Question 7

SLPaper 1

A botanical garden tracks rare orchid blooms, where blooms occur according to a Poisson process with a mean rate of 1.5 blooms per week. Let $U$ be the number of blooms in a week, and let $V$ be the number of blooms photographed, with only the first two blooms per week photographed due to limited staff. Additionally, a visitor rolls a biased six-sided die with faces labeled 1 to 6 to determine the number of bloom photos they view, with probability distribution given by:

$w$	1	2	3	4	5	6
$P(W=w)$	$p$	$p$	$2 p$	0.2	$q$	0.1

The garden's promotional campaign requires that the probability of a visitor viewing at least as many photos as the number of photographed blooms ( $W \geq V$ ) is exactly 0.5 .

Find the values of $p$ and $q$ .

[4]

Construct the probability distribution table for $V$ and sketch it for $v=0,1,2$ .

[4]

Find the expected number of photographed blooms, $E(V)$ .

[2]

Determine the values of $p$ and $q$ such that $P(W \geq V)=0.5$ .

[5]

Question 8

SLPaper 1

A discrete random variable $Z$ represents the number of defective items in a batch, with the following probability distribution:

$z$	0	1	2	3
$P(Z=z)$	$\frac{1}{6}$	$\frac{1}{3}$	$p$	$q$

Given that $E\left(Z^{2}\right)=\frac{19}{6}$ ,

find the values of $p$ and $q$ .

[4]

find the standard deviation of $Z$ .

[3]

Question 9

SLPaper 2

A tech company tests microchips for defects, with each chip having a 20

Show that $P(Z=3)=0.008$ and find $P(Z=4)$ .

[2]

Find $E(Z)$ and sketch the probability distribution for $n=3,4,5,6$ .

[4]

Prove that the probability distribution has a single mode at $n=8$ .

[5]

Find the variance of $Z$ .

[4]

Question 10

SLPaper 1

A city's traffic monitoring system records vehicles passing through a busy intersection. Let $X$ be the discrete random variable representing the number of electric vehicles (EVs) passing per hour, modeled by a Poisson distribution with mean 3.2. Due to limited charging stations nearby, only the first three EVs each hour are recorded as "serviced" (able to charge). Let $Y$ be the number of serviced EVs per hour. The city uses this data to plan infrastructure upgrades, but the system's accuracy is affected by occasional sensor failures, which occur independently with probability 0.05 per hour.

Find the probability that exactly two EVs pass the intersection in a given hour.

[2]

Construct the probability distribution table for $Y$ .

[4]

Calculate the expected number of serviced EVs per hour, $E(Y)$ .

[2]

In a 24 -hour period, find the probability that there are at most 5 hours with sensor failures.

[3]

w

P(W=w)

0.2

p

0.3 p

x

P(X=x)

\frac{1}{5}

p

\frac{1}{10}

q

z

P(Z=z)

k

0.2

0.1

0.008

a

w

P(W=w)

p

p

2 p

0.2

q

0.1

z

P(Z=z)

\frac{1}{6}

\frac{1}{3}

p

q