Discrete random variables

A uniform object is formed by connecting three solid cubes with edge lengths $3\,\text{m}$, $2\,\text{m}$ and $1\,\text{m}$. The object has an axis of symmetry, and the cubes are arranged one above another with the cube of edge $2\,\text{m}$ positioned between the other two cubes (see diagram).

Feb/March 2019

A fair ordinary die is rolled over and over until either a 1 or a 6 appears.

Feb/March 2020

A fair spinner with $5$ sides labelled $1, 2, 3, 4, 5$ is spun many times. Each spin gives the score shown by the side where the spinner stops.

Feb/March 2021

The random variable $X$ can take only the values 1, 2, 3, 4. The probability that $X$ has the value $x$ is $kx(5-x)$, where $k$ is a constant.

Feb/March 2021

A fair red spinner has edges labelled $1, 2, 2, 3$. A fair blue spinner has edges labelled $-3, -2, -1, -1$. Each spinner is spun once, and the number shown on the edge where each spinner stops is recorded. The random variable $X$ represents the total of the two numbers obtained.

Feb/March 2022

In a particular country, the chance of getting more than $10\,\text{cm}$ of rain on any one day is $0.18$, independently of the weather on any other day.

Feb/March 2022

Alisha owns four coins. One of them is biased so that the probability of getting a head is $0.6$. The other three coins are fair. Alisha tosses all four coins at the same time. The random variable $X$ represents the number of heads obtained.

Feb/March 2023

Eighty percent of the people living in Kinwawa support the idea of a leisure centre being constructed in the town. Twenty Kinwawa residents are picked at random and, one by one, are asked whether they are in favour of the leisure centre.

Feb/March 2023

Sam belongs to a soccer club and is practising how to score goals. On each attempt, the probability that Sam scores a goal is $0.7$, independent of every other attempt.

Feb/March 2024

Anil is competing in a tournament. In each match of this tournament, a player earns $2$ points for a win, $1$ point for a draw and $0$ points for a loss. For any of Anil’s games, the probabilities that he will win, draw or lose are $0.5$, $0.3$ and $0.2$ respectively. The outcomes of the games are mutually independent. The random variable $X$ represents the total number of points that Anil obtains from his first $3$ games in the tournament.

Feb/March 2024

Jacob flips three coins at the same time. The first coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{3}$. The second coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{4}$. The third coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{5}$. Let the random variable $X$ represent the total number of heads obtained.

Feb/March 2025

Particle $P$ has mass $0.2\,\text{kg}$ and is fastened to one end of a light inextensible string of length $0.6\,\text{m}$. The free end of that string is fixed at point $A$. $P$ is also attached to one end of a second light inextensible string of length $0.6\,\text{m}$, and the other end is fixed at point $B$ directly below. The particle travels in a horizontal circle of radius $0.3\,\text{m}$, centred at the midpoint of $AB$, with both strings remaining straight (see diagram).

May/June 2018

A light elastic string has natural length $a\text{ m}$ and modulus of elasticity $\lambda\text{ N}$. Its tension is $4\text{ N}$ when the length is $1.6\text{ m}$, and it is $6\text{ N}$ when the length is $2\text{ m}$. One end of the string is fixed at a point $O$ on a smooth horizontal surface, while the other end is connected to a particle $P$ of mass $0.2\text{ kg}$. The particle $P$ moves at constant speed on the surface in a circle with centre $O$ and radius $1.9\text{ m}$.

May/June 2019

For two fair six-sided dice, the score is the total of the numbers showing on the top faces. The dice are rolled again and again until a score of 4 is obtained. Let the random variable $X$ denote the number of throws needed.

May/June 2020

A firm makes small sweet boxes containing 5 jellies and 3 chocolates. Jemeel randomly picks 3 sweets from one box.

May/June 2020

One fair three-sided spinner is labelled 1, 2, 3. Another fair five-sided spinner is labelled 1, 1, 2, 2, 3. Each spinner is spun once, and the number showing on the side where it lands is recorded. The random variable $X$ is the larger of the two numbers when they are different, and their shared value when they are the same.

May/June 2020

On any one day, the chance that Moena sends a message to her friend Pasha is $0.72$.

May/June 2020

At one large college, $22\%$ of the students have a car.

May/June 2020

A fair four-sided spinner is labelled on its edges 1, 2, 2, 3. A fair three-sided spinner is labelled on its edges $-2$, $-1$, 1. Each spinner is spun, and the number on the edge at which it stops is recorded. The random variable $X$ is the total of the two recorded numbers.

May/June 2020

A pair of fair coins is tossed over and over until a pair of tails appears. The random variable $X$ represents the number of throws needed to obtain a pair of tails.

May/June 2020

In Questa, $60\%$ of adults go to work by car.

May/June 2021

Sharma is aware that her cupboard contains 3 tins of carrots, 2 tins of peas and 2 tins of sweetcorn. Every tin has the same shape and size, but all the labels have been removed, so Sharma cannot tell what is in any tin. Sharma wants carrots for her meal, so she opens the tins one at a time, chosen at random, until she opens a tin of carrots. The random variable $X$ represents the number of tins she has to open.

May/June 2021

An ordinary fair die is thrown again and again until a 5 appears. Let the random variable $X$ represent the number of throws needed.

May/June 2021

Two fair spinners are used. One has faces labelled $1, 2, 2$ and the other has faces labelled $-2, 0, 1$. Each spinner is spun once, and the number showing when it stops is recorded. The random variable $X$ represents the total of the two numbers.

May/June 2021

Each day, Richard boards a flight from Astan to Bejin. For any one day, the chance that the flight is early is $0.15$, the chance that it is on time is $0.55$ and the chance that it is late is $0.3$.

May/June 2021

The random variable $X$ is restricted to the five values $-2$, $-1$, $0$, $1$, $2$. Its probability distribution is shown in this table: $P(X=-2)=p$, $P(X=-1)=p$, $P(X=0)=0.1$, $P(X=1)=q$, $P(X=2)=q$.

May/June 2021

In Arka, the three villages Reeta, Shan and Teber contain 800 households in total. Every household was questioned about the standard of its broadband service. The results are shown in the table below.

May/June 2021

A balanced 6-sided die has the faces marked $1, 2, 2, 3, 3, 3$. It is thrown twice. The random variable $X$ stands for the sum of the two numbers shown.

May/June 2022

The random variable $X$ can take the values $-2, 1, 2, 3$. You are told that $P(X = x) = kx^2$, where $k$ is a constant.

May/June 2022

Ramesh rolls a standard fair die with 6 sides.

May/June 2022

Eli has four fair $4$-sided dice, each face labelled $1, 2, 3, 4$. He rolls all four together. The random variable $X$ stands for the number of $2$s that appear.

May/June 2023

A wildlife magazine for children is issued every Monday. During the next 12 weeks, each edition will come with a model animal as a free gift. The five possible models are tiger, leopard, rhinoceros, elephant and buffalo, and each has the same chance of being placed in the magazine. Sahim purchases one copy of the magazine every Monday.

May/June 2023

The random variable $X$ may take the values $-2$, $2$ and $3$. You are told that $P(X = x) = k(x^2 - 1)$, where $k$ is a constant.

May/June 2023

A fair 5-sided spinner is numbered $1, 2, 3, 4, 5$ on its faces. It is spun again and again until the side showing $2$ appears on the face where the spinner lands. The random variable $X$ represents how many spins are needed.

May/June 2023

Two fair coins are tossed together again and again until a pair of heads appears. Let the random variable $X$ stand for the number of tosses needed.

May/June 2023

The random variable $X$ can be $1, 2, 3, 4$. It is stated that $P(X = x) = kx(x + a)$, with $k$ and $a$ as constants.

May/June 2023

Harry has three coins. One of the coins is biased, with the chance of getting a head on a throw equal to $\frac{1}{3}$. The second coin is biased, with the chance of getting a head on a throw equal to $\frac{1}{4}$. The third coin is biased, with the chance of getting a head on a throw equal to $\frac{1}{5}$. Harry throws all three coins. The random variable $X$ counts the number of heads he gets.

May/June 2024

Each year, Rajesh enters once for a ticket to a music festival. His chance of success in any given year is $0.3$, and the outcomes in different years are independent.

May/June 2024

Jasmine owns one $5 coin, two $2 coins and two $1 coins. She picks two of these coins at random. The random variable $X$ gives the total value, in dollars, of the two chosen coins.

May/June 2024

Residents of Mahjing were surveyed and asked to sort their local bus service: 25% of residents described the service as good. 60% of residents described the service as satisfactory. 15% of residents described the service as poor.

May/June 2024

A fair six-sided die has faces numbered 1, 2, 2, 3, 3, 3. The random variable $X$ gives the total score after the die is rolled twice.

May/June 2024

Every day, Salah chooses to try the crossword puzzle in his newspaper. On any one day, the chance that he completes the puzzle is $0.65$, and the days are independent of each other.

May/June 2024

A bag has 10 marbles: 4 are red and 6 are blue. Four marbles are drawn at random from the bag without replacement. The random variable $X$ represents the number of blue marbles drawn.

May/June 2025

Rachel owns three coins. The first coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{3}$. The second coin is biased so that the probability of getting a head when it is tossed is $\frac{1}{4}$. The third coin is fair. Rachel tosses all three coins at the same time. The random variable $X$ represents the number of tails she gets.

May/June 2025

At a particular road junction from Bromley, vehicles must turn left, right or continue straight ahead. It is known over time that 30% turn left, 25% turn right and 45% continue straight ahead. The driver of every vehicle makes the choice independently of every other driver.

May/June 2025

A fair six-sided dice whose faces are numbered $1, 2, 3, 4, 5, 6$ is rolled repeatedly until a $3$ appears. The number of rolls required is represented by the random variable $X$.

May/June 2025

Two fair 6-sided dice labelled with faces $1, 2, 3, 4, 5, 6$ are tossed. The two results are recorded. The random variable $X$ is defined as follows: if the two results are the same, $X = 0$. If the results are different, $X$ is the larger result minus the smaller result.

May/June 2025

A light elastic string has natural length $2\,\text{m}$ and modulus of elasticity $\lambda\,\text{N}$. Its ends are fastened to fixed points $A$ and $B$, which lie at the same horizontal level and are $2.4\,\text{m}$ apart. A particle $P$ of mass $0.6\,\text{kg}$ is attached to the midpoint of the string and hangs in equilibrium at a point $0.5\,\text{m}$ below $AB$ (see diagram).

Oct/Nov 2010

Kayla is taking part in a throwing event. A throw counts as a success when the distance thrown is more than $30$ metres. The probability that Kayla gets a success on any throw is $0.25$.

Oct/Nov 2020

The random variable $X$ can take the values $1$, $2$, $3$ and $4$, with each value occurring with probability $\frac{1}{4}$. Two independent values of $X$ are selected at random. If the two selected values of $X$ are identical, then the random variable $Y$ is that value. If they are not identical, then $Y$ is the larger value of $X$ minus the smaller value of $X$.

Oct/Nov 2020

A fair die with six faces labelled $1, 2, 3, 4, 5, 6$ is rolled again and again until a $4$ appears.

Oct/Nov 2020

A bag has $5$ red balls and $3$ blue balls inside it. Sadie selects $3$ balls at random from the bag, without replacement. The random variable $X$ denotes the number of red balls that she selects.

Oct/Nov 2020

An ordinary fair die is rolled repeatedly until a $6$ appears.

Oct/Nov 2020

Each of the three coins $A$, $B$ and $C$ is tossed once. Coins $A$ and $B$ are both biased so that the chance of a head is $\frac{2}{3}$. Coin $C$ is biased so that the chance of a head is $\frac{4}{5}$. The random variable $X$ counts how many heads are obtained when the three coins are tossed.

Oct/Nov 2020

Two fair coins are tossed together. The random variable $X$ represents the number of paired tosses needed until two tails are seen at the same time for the first time.

Oct/Nov 2021

One fair spinner has edges labelled $0, 1, 2, 2$. A second fair spinner has edges labelled $-1, 0, 1$. Both spinners are spun once. For each spinner, the number on the edge where it stops is recorded. The random variable $X$ represents the total of the two numbers.

Oct/Nov 2021

The bag has $5$ yellow marbles and $4$ green marbles. Three marbles are chosen at random from the bag, without replacement.

Oct/Nov 2021

In a game, Jim throws three darts at a board; this counts as one ‘turn’. The board’s centre is known as the bull’s-eye. Let the random variable $X$ represent how many darts in a turn land on the bull’s-eye. The probability distribution of $X$ is shown in the table below: $x = 0, 1, 2, 3$ with $P(X = x) = 0.6, p, q, 0.05$. You are told that $\text{E}(X) = 0.55$.

Oct/Nov 2021

The probability distribution of a random variable $X$ is given in the table below.

Oct/Nov 2022

Three fair $6$-sided dice, with faces labelled $1, 2, 3, 4, 5, 6$, are thrown together repeatedly. For each throw, the score is the total of the numbers showing on the upper faces.

Oct/Nov 2022

Eric has three coins. One coin is fair, while each of the other two is biased so that the chance of getting a head on any throw is $\frac{1}{4}$, independently of every other throw. Eric throws all three coins at the same time. Events $A$ and $B$ are defined below. $A$: all three coins show matching outcomes $B$: at least one of the biased coins shows a head

Oct/Nov 2022

There are three fair 4-sided spinners, each labelled $1, 2, 3, 4$ on its sides. They are spun together, and the number showing on each spinner is noted. The random variable $X$ is the largest number recorded.

Oct/Nov 2022

Hazeem keeps throwing two standard fair 6-sided dice together. Each time, the result is the total of the two numbers she gets.

Oct/Nov 2023

In a throwing competition, a competitor gets three attempts to throw a ball as far as possible. The random variable $X$ represents how many throws go beyond $30$ metres. The probability distribution table for $X$ is given below.

Oct/Nov 2023

George uses a fair $5$-sided spinner whose faces are labelled $1, 2, 3, 4, 5$. He spins it and records the number shown on the face where it comes to rest.

Oct/Nov 2023

Becky works in an office at some times and from home at other times. The random variable $X$ represents the number of days in any one week that she works at home. It is given that $\text{P}(X = x) = kx(x + 1)$, where $k$ is a constant and $x = 1, 2, 3$ or $4$ only.

Oct/Nov 2023

On any one attempt, the chance that a driver succeeds in an advanced driving test is $0.3$.

Oct/Nov 2023

Nicola rolls a fair ordinary six-sided dice. The random variable $X$ represents how many throws she needs to obtain a 6.

Oct/Nov 2024

The random variable $X$ may assume the values $-2, -1, 0, 2, 3$. It is stated that $P(X = x) = k(x^2 + 2)$, where $k$ is a positive constant.

Oct/Nov 2024

In a college, each student selects just one sport from tennis, hockey or netball. The table gives the numbers of Year 1 and Year 2 students at the college who play each sport. One student is selected at random from the 120 students. The events $X$ and $N$ are defined as follows: $X$: the student is in Year 1. $N$: the student plays netball.

Oct/Nov 2024

A fair coin and a fair ordinary six-sided die are tossed together. The random variable $X$ is defined like this: when the coin lands tails, $X$ is double the die score; when the coin lands heads, $X$ equals the die score if it is even, and $X$ is $0$ otherwise.

Oct/Nov 2024

In a game, a player tries to score by sending a ball into the net. The probability that Leno scores a goal is $0.4$ on any attempt, independently of all other attempts. The random variable $X$ represents how many attempts Leno needs before he scores a goal.

Oct/Nov 2024

A fair red six-sided dice has faces labelled $1, 1, 1, 2, 2, 2$. A fair blue six-sided dice has faces labelled $1, 1, 2, 2, 3, 3$. Both dice are thrown. The random variable $X$ represents the product of the two scores.

Oct/Nov 2024

The random variable $X$ can take the value $x$ with probability $kx^2$, where $k$ is a constant, and the only possible values of $x$ are $-2$, $1$, $2$ and $3$.

Oct/Nov 2025

On any day, Cooper either puts on a blue jumper, puts on a green jumper, or does not put on a jumper. The probability that he puts on a blue jumper is $0.6$ and the probability that he puts on a green jumper is $0.3$. What Cooper wears, or does not wear, on one day is independent of his decision on any other day.

Oct/Nov 2025

A coin is loaded so that the chance of a head on any throw is 0.4. The coin keeps being thrown until the first head comes up.

Oct/Nov 2025

Kayla has a bag that contains 3 red marbles, 1 blue marble and 2 green marbles. She chooses one marble at random from the bag and does not put it back. She keeps repeating this procedure until a green marble is obtained. The random variable $X$ is the number of marbles she must choose before obtaining a green marble.

Oct/Nov 2025

A fair red spinner has 4 sides labelled 1, 2, 3, 4. A fair blue spinner has 4 sides labelled 0, 1, 2, 3. When a spinner is spun, the score is the number shown on the side where it comes to rest. Both spinners are spun at the same time. The random variable $X$ represents the larger of the two scores obtained. If the two scores are the same, then $X$ takes the value 0.

Oct/Nov 2025

A fair six-sided die with faces numbered 1, 2, 3, 4, 5, 6 is thrown repeatedly. The random variable $X$ represents the number of throws needed before a 6 is obtained.

Oct/Nov 2025

Kai owns a spinner with four sides, marked 1, 2, 3, 4. Each time it is spun, the score equals the number on the face where it stops. The random variable $X$ represents this score. The probability distribution table for $X$ appears below.

Oct/Nov 2025