The Poisson distribution

A flower shop contains $5$ yellow roses, $3$ red roses and $2$ white roses. Martin picks $3$ roses at random.

Feb/March 2016

In one town, $35\%$ of residents take a holiday abroad, while $65\%$ spend their holiday in their own country. Among those who go abroad, $80\%$ choose the seaside, $15\%$ go camping and $5\%$ take a city break. Among those who have a holiday in their own country, $20\%$ go to the seaside and the remaining people are split equally between camping and a city break.

Feb/March 2016

A bag has $10$ pink balloons, $9$ yellow balloons, $12$ green balloons and $9$ white balloons. $7$ balloons are chosen at random without replacement.

Feb/March 2017

It is given that $10\%$ of the population enjoy watching Historical Drama on television.

Feb/March 2017

Whenever Tamar goes to work, he either wears a blue suit with probability $0.6$ or a grey suit with probability $0.4$. If his suit is blue, the probability that he wears red socks is $0.2$. If his suit is grey, the probability that he wears red socks is $0.32$.

Feb/March 2019

The random variable $X$ can only have the values $-1, 1, 2, 3$. The probability that $X$ takes the value $x$ is $kx^2$, where $k$ is a constant.

Feb/March 2019

A survey carried out by a large supermarket found that $35\%$ of its customers buy online.

Feb/March 2019

On average, the booklets from a certain publisher have 1 incorrect letter in every 30 letters, and the mistakes arise at random. A booklet chosen at random from this publisher has 12500 letters.

Feb/March 2020

The count of accidents on one particular road follows a Poisson distribution with mean $0.4$ per $50$-day period.

Feb/March 2020

At a particular factory, $1$ in $400$ microchips produced are faulty on average. Let $X$ stand for the number of faulty microchips in a random sample of $1000$.

Feb/March 2021

Ponds $A$ and $B$ each hold a very large population of fish. It is known that $2.4\%$ of the fish in pond $A$ are carp, while $1.8\%$ of the fish in pond $B$ are carp. Random samples of $50$ fish from pond $A$ and $60$ fish from pond $B$ are taken. Apply suitable Poisson approximations to determine the following probabilities.

Feb/March 2022

Over an 8-hour working day, the count of orders reaching a shop is represented by the random variable $X$ with distribution $\text{Po}(25.2)$.

Feb/March 2023

In this lottery, the average is that 1 in every 10,000 tickets is a prize-winning ticket, and an agent sells 6000 tickets.

Feb/March 2024

On any day, the numbers of girls and boys who are late for her class are represented by the independent random variables $G \sim \text{Po}(0.10)$ and $B \sim \text{Po}(0.15)$, respectively.

Feb/March 2024

Throughout the holidays, Martin spends $25\%$ of each day playing computer games, and his friend rings him once per day at a time chosen at random.

May/June 2010

In a television quiz show, Peter tackles questions in order and the game stops immediately once a question is answered incorrectly. The probabilities are: Peter answers correctly himself $0.7$; Peter answers wrongly himself $0.1$; Peter decides to ask for help $0.2$. The first time Peter chooses to ask for help, he asks the audience. The probability that the audience supplies the correct answer is $0.95$.

May/June 2010

A bottle holds 13 red sweets, 13 blue sweets, 13 green sweets and 13 yellow sweets. 7 sweets are chosen at random.

May/June 2010

Every day, Christa walks her dog. On any day, the chance that they visit the park is $0.6$. When they do go to the park, the probability that the dog barks is $0.35$. When they do not go to the park, the probability that the dog barks is $0.75$.

May/June 2010

Biscuits are packed in groups of 18. Each biscuit has a fixed chance of being broken, independent of all the others. The average number of broken biscuits per packet is $2.7$.

May/June 2011

While Ted is trying to locate his pen, the chance that it is inside his pencil case is $0.7$. If the pen is in his pencil case, he always succeeds in finding it. If the pen is elsewhere, the chance that he finds it is $0.2$.

May/June 2011

Determine the probability of obtaining at least one $3$ when 9 fair dice are thrown.

May/June 2011

A loaded die was rolled 20 times, and the number of 5s was recorded. The trial was carried out many times, and the mean number of 5s was determined to be $4.8$.

May/June 2011

Judy and Steve take part in a game with five cards labelled $3, 4, 5, 8, 9$. Judy first picks one card at random, reads its number and puts the card back. Steve then does the same, choosing a card at random, reading the number and replacing the card. If the two numbers match, the score is $0$. If they are different, the smaller number is taken away from the larger number to give the score. When the score is $0$, they take another turn. When the score is $4$ or more, Judy wins; otherwise Steve wins. They keep playing until one player wins.

May/June 2011

The chance that Sue finishes a Sudoku puzzle correctly is $0.75$.

May/June 2011

Maria has $3$ stations already stored on her radio. When she turns the radio on, there is a probability of $0.3$ that it will tune to station $1$, a probability of $0.45$ that it will tune to station $2$, and a probability of $0.25$ that it will tune to station $3$. On station $1$, the probability that the presenter is male is $0.1$; on station $2$, the probability that the presenter is male is $0.85$; and on station $3$, the probability that the presenter is male is $p$. When Maria switches on the radio, the probability that it is on station $3$ with a male presenter is $0.075$.

May/June 2012

During winter in one mountainous area, the chance that more than $20\text{ cm}$ of snow falls on a given day is $0.21$.

May/June 2012

At Restaurant Bijoux, 13% of customers judged the food to be ‘poor’, 22% judged it to be ‘satisfactory’ and 65% judged it to be ‘good’. A random sample of 12 customers who had a meal at Restaurant Bijoux was selected.

May/June 2012

For a person chosen at random, the weekday of their next birthday is equally likely to be any day of the week, and it is independent of every other person’s birthday.

May/June 2013

$Q$ is the event "Nicola throws two fair dice and obtains a total of $5$". $S$ is the event "Nicola throws two fair dice and gets one low score $(1, 2 \text{ or } 3)$ together with one high score $(4, 5 \text{ or } 6)$".

May/June 2013

In a particular country, the average is that one student in five has blue eyes.

May/June 2013

State three conditions that have to be met for a situation to be modelled by a binomial distribution.

May/June 2014

In a particular country, $12\%$ of houses have solar heating, and $19$ houses are selected at random.

May/June 2014

Determine how many different numbers can be formed from some or all of the digits of the number $1\,345\,789$ if

May/June 2014

Tom and Ben keep playing the game again and again. The chance that Tom wins a single game is 0.3. Every game is won by exactly one of Tom or Ben. They stop once either player, called the champion, has won two games.

May/June 2014

Jason rolls two fair dice, each with faces labelled $1$ to $6$. Event $A$ means ‘exactly one of the numbers obtained is divisible by $3$, while the other is not divisible by $3$’. Event $B$ means ‘the product of the two numbers obtained is even’.

May/June 2015

A survey is carried out to discover how many photos people capture on a one-week holiday and how often they look back at earlier photos. For a randomly chosen person, the probability of taking fewer than $100$ photos is $x$. The probability that these people view past photos at least $3$ times is $0.76$. For those who take at least $100$ photos, the probability that they view past photos fewer than $3$ times is $0.90$. This information is shown in the tree diagram. The probability that a randomly chosen person views past photos fewer than $3$ times is $0.801$.

May/June 2015

A fair die is rolled $10$ times. Determine the probability that the number of sixes rolled is between $3$ and $5$ inclusive.

May/June 2015

Nikita is shopping for a birthday present for her mother. She chooses a scarf with probability $0.3$, otherwise she chooses a handbag. The chance that her mother likes the scarf is $0.72$. The chance that her mother likes the handbag is $x$. The tree diagram shows this information. The probability that Nikita’s mother likes the present Nikita buys is $0.783$.

May/June 2015

The masses, in grams, of onions in a supermarket are normally distributed with mean $\mu$ and standard deviation $22$. The probability that one randomly selected onion has a mass above $195$ grams is $0.128$.

May/June 2015

For cameras on a production line, the chance that a camera chosen at random is substandard is $0.072$. A random sample of $300$ cameras is inspected.

May/June 2015

The six faces of a biased die carry the labels $1,2,3,4,5$ and $6$. Let the random variable $X$ denote the score obtained when the die is tossed. The probability distribution for $X$ is as follows: for $x=1,2,3,4$, $P(X=x)=p$, and for $x=5,6$, $P(X=x)=0.2$.

May/June 2016

On average, among people who enter this large shop, 34% leave without buying anything, 53% spend less than $50, and 13% spend at least $50.

May/June 2016

A group of passengers is making the journey to Picton in a minibus. For each passenger, the probability of carrying a backpack is $0.65$, independent of the others. Each minibus has $12$ seats for passengers.

May/June 2016

Ashfaq rolls two fair dice and records the results. $R$ is the event ‘The product of the two numbers is $12$’. $T$ is the event ‘Exactly one of the numbers is odd and the other is even’.

May/June 2017

Every week, the Redbury United soccer team takes part in one match. Each match results in a win, a draw or a loss. At the beginning of the soccer season, the probability that Redbury United win their first match is $\frac{3}{5}$, and the probabilities of losing or drawing are equal. If they win the first match, the probability that they win the second match is $\frac{7}{10}$ and the probability that they lose the second match is $\frac{1}{10}$. If they draw the first match they are equally likely to win, draw or lose the second match. If they lose the first match, the probability that they win the second match is $\frac{3}{10}$ and the probability that they draw the second match is $\frac{1}{20}$.

May/June 2017

Eggs come packed in boxes of 20, and cracked eggs occur independently; the average number of cracked eggs in a box is $1.4$.

May/June 2017

Hebe works on a crossword puzzle each day. Let $X$ stand for how many puzzles she finishes in a week (7 days). On average, Hebe finishes $7$ out of $10$ of these puzzles.

May/June 2017

Vehicles arriving at a particular road junction from town A may turn left, turn right, or continue straight ahead. It has been observed over time that, among vehicles approaching this junction from town A, $55\%$ turn left, $15\%$ turn right and $30\%$ continue straight ahead. The direction taken by one vehicle at the junction is independent of the direction chosen by any other vehicle at the junction.

May/June 2018

In any given week, the chance that Janice buys one item online is $0.35$. In a single week, Janice never purchases more than one online item.

May/June 2019

Megan communicates with her friends in one of $3$ possible ways: text, email or social media. For each message, the chance that she uses text is $0.3$ and the chance that she uses email is $0.2$. She gets an immediate reply to a text message with probability $0.4$, to an email with probability $0.15$ and to social media with probability $0.6$.

May/June 2019

Mr and Mrs Keene and their $5$ children attend a football match with their friends Mr and Mrs Uzuma and their $2$ children. Determine the number of different ways the $11$ people can form a line at the entrance in each of the situations below.

May/June 2019

In each week, a sports team plays one home match and one away match. For their home fixtures, they score goals at a constant average rate of $2.1$ goals per match. For their away fixtures, they score goals at a constant average rate of $0.8$ goals per match. You may assume that goals are scored at random times and independently of one another.

May/June 2020

In a firm’s data-entry department, it is known that $0.12\%$ of data items are recorded incorrectly, and that these errors arise at random and independently.

May/June 2020

The random variable $X$ follows the distribution $\text{Po}(\lambda)$.

May/June 2020

The random variable $A$ follows the distribution $\text{Po}(1.5)$. $A_1$ and $A_2$ are independent observations of $A$.

May/June 2020

Accidents at the two factories happen independently and at random. The average monthly accident counts are $3.1$ for factory $A$ and $1.7$ for factory $B$.

May/June 2021

The graph of the probability density function for the random variable $X$ is symmetric around the line $x = 4$.

May/June 2021

On average, $1$ adult out of every $75\,000$ has a particular genetic disorder.

May/June 2021

The random variable $X$ follows the distribution $B(400,\,0.01)$.

May/June 2021

Customers turn up at a certain shop at random times. It has been established that the average number of customers arriving in a $5$-minute period is $2.1$.

May/June 2021

A certain kind of plant usually has three leaves. However, it is known that, on average, $1$ in $10\,000$ of these plants has four leaves, and plants with four leaves are described as ‘lucky’. Let $X$ represent the number of lucky plants in a random sample of $25\,000$ plants.

May/June 2021

Cars reach a fuel station randomly, with a steady mean arrival rate of 13.5 per hour.

May/June 2022

It is stated that $1.8\%$ of the children in one country have not had measles vaccination. A simple random sample of $200$ children from this country is selected.

May/June 2022

The count of clients reaching an information desk is modelled by a Poisson distribution with mean $2.2$ in each $5$-minute interval.

May/June 2022

In one country, 20540 adults from a population of 601200 hold a degree in medicine.

May/June 2023

The random variable $W$ follows a Poisson distribution. State the relationship between $\mathrm{E}(W)$ and $\mathrm{Var}(W)$.

May/June 2023

The random variable $X$ gives the number of books taken in by a charity shop each day, and its average rate is fixed at $5.1$ per day.

May/June 2023

It is given that 1 in 5000 people in Atalia has this condition. A random sample of 122500 people from Atalia is selected for a medical trial. Let $X$ represent the number who have the condition.

May/June 2023

The bus station has four entrances exactly. In the morning, the passenger counts arriving at these entrances in a 10-second interval follow the independent distributions $\text{Po}(0.4)$, $\text{Po}(0.1)$, $\text{Po}(0.2)$ and $\text{Po}(0.5)$.

May/June 2024

The sales of cell phones at this shop happen one at a time, at random and independently. The mean number sold per hour is $1.2$. Assume now that a Poisson distribution gives an appropriate model.

May/June 2024

A random variable $X$ follows the distribution $\mathrm{Po}(145)$.

May/June 2024

The number of goals scored by a sports team during the first half of a match is modelled by $X \sim \text{Po}(3.1)$. The number of goals scored by the same team during the second half of a match is modelled by $Y \sim \text{Po}(2.4)$. You may assume that the distributions of $X$ and $Y$ are independent.

May/June 2024

The random variable $X$ follows the distribution $B(4000,\,0.001)$.

May/June 2024

It is established that $1\%$ of the houses in one area are fitted with a wind turbine. For a survey on domestic heating, a random sample of $400$ houses from this area is selected. Let $X$ represent how many houses in the sample have a wind turbine.

May/June 2025

The random variables $W$ and $X$ are independent, with distributions $\text{Po}(1.2)$ and $\text{Po}(2.3)$ respectively.

May/June 2025

The random variable $X$ follows the distribution $\text{Po}(15)$.

May/June 2025

Answer the following using suitable approximating distributions.

May/June 2025

In one shop, customers come independently and at random, with a steady average rate of $23.4$ per hour.

May/June 2025

$X$ is a random variable with distribution $\text{Po}(1.5)$. Let $S$ denote the total of three independent values of $X$.

May/June 2025

The numbers of cars and trucks per minute arriving at a fuel station are represented by independent variables with distributions $\text{Po}(0.8)$ and $\text{Po}(0.5)$, respectively.

May/June 2025

On average, 2 apples in every 15 are considered to be underweight.

Oct/Nov 2010

Three friends, Rick, Brenda and Ali, attend a football match but do not agree on which entrance to the ground they will use to meet. The ground has four entrances, $A$, $B$, $C$ and $D$. Each friend selects an entrance independently.

Oct/Nov 2010

The discrete random variable $X$ can take only the values $1, 4, 5, 7$ and $9$. The table shows the probability distribution of $X$.

Oct/Nov 2010

The spinner is fair and has five sides labelled $1, 2, 3, 4, 5$. Raj spins it and then throws two fair dice. His score is determined as follows: when the spinner lands on an even-numbered side, Raj multiplies the two dice results to obtain his score. When the spinner lands on an odd-numbered side, Raj adds the two dice results to obtain his score.

Oct/Nov 2010

For any given day, the chance that Julie’s train arrives late is $0.3$.

Oct/Nov 2010

In this probability distribution, the random variable $X$ may only take the values $1, 2, 3, 4, 5$, and the probability of getting $x$ is $kx$.

Oct/Nov 2010

An aeroplane has $14$ passenger seats. They are laid out as $4$ rows of $3$ seats, with a final back row containing $2$ seats (see diagram). $12$ passengers board the plane.

Oct/Nov 2010

There are twelve coins thrown and laid out in a row, and each one may display either a head or a tail.

Oct/Nov 2011

A triangular spinner has one red face, one blue face and one green face. The red face is weighted so that the chance of landing on red is four times the chance of landing on blue. The green face is weighted so that the spinner is three times as likely to stop on green as on blue.

Oct/Nov 2011

A company staged a fireworks display made up of 20 fireworks. For each firework, the chance that it does not work is $0.05$, independently of the others. Each of the 20 fireworks costs the company $24$. 450 people pay the company $10$ each to watch the display. If more than 1 firework fails to work they get their money back.

Oct/Nov 2012

Ana sees her friends every day. On any given day, the chance that she arrives early is $0.05$, while the chance that she arrives late is $0.75$. In all other cases, she is on time.

Oct/Nov 2012

A chess team consisting of $2$ girls and $2$ boys is to be selected from the $7$ girls and $6$ boys in the chess club. Determine how many selections are possible if $2$ of the girls are twins and they must either both be included or both be excluded.

Oct/Nov 2012

On Saturday afternoons, Mohit either shops with probability $0.25$, goes to the cinema with probability $0.35$, or stays at home. If he shops, the probability that he spends more than $50 is $0.7$. If he goes to the cinema, the probability that he spends more than $50 is $0.8$. If he remains at home, he spends $10 on a pizza.

Oct/Nov 2013

During the morning rush hour on trains, each person is classified as a student with probability $0.36$, an office worker with probability $0.22$, a shop assistant with probability $0.29$, or none of these.

Oct/Nov 2013

For a large consignment of mangoes, $15\%$ are put in the small group, $70\%$ in the medium group and $15\%$ in the large group.

Oct/Nov 2013

Sarah gets $X$ phone calls per day, and the probability distribution is given below.

Oct/Nov 2014

Packets contain $15$ screws each. Faulty screws arise at random. A large sample of packets is checked for faulty screws, and the average number of faulty screws in one packet is $1.2$.

Oct/Nov 2014

Sharik tries an online multiple-choice revision question. There are $3$ suggested answers, and exactly one is correct. When Sharik selects an answer, the computer shows whether it is right or wrong. First, Sharik picks one of the three suggested answers at random. If that choice is wrong, he gets a second attempt by selecting at random from the other $2$. If that too is wrong, Sharik then selects the last remaining answer, which has to be correct.

Oct/Nov 2014

In each year, the number of books read by members of a book club has the binomial distribution $B(12,\,0.7)$.

Oct/Nov 2014

Seven fair dice, each labelled $1,\,2,\,3,\,4,\,5,\,6$, are rolled and arranged in a row. Find how many arrangements are possible if the two end numbers have a total of $4$.

Oct/Nov 2014