DP IB Maths: AI HL

Topic Questions

4.10 Poisson Distribution

1a
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3 marks

State which distribution – normal, binomial or Poisson – is likely to be appropriate for calculating the final value of each of the following probabilities.  In each case specify any assumptions that would need to be made, and any parameters of which you would need to know the value in order to carry out the calculation.

The probability that at least 7 college students in a class of 30 receive a text message in the next hour.

1b
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3 marks

The probability that a randomly selected college student in a class receives at least 7 text messages in the next hour.

1c
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3 marks

The probability that a randomly selected college student takes less than 5 minutes to write three random text messages.

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2a
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1 mark

The random variable X follows a Poisson distribution which has a standard deviation of 2.25.

Write down the value for straight E open parentheses X close parentheses.

2b
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2 marks

Find straight P open parentheses X equals 4 close parentheses.

2c
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3 marks

Find straight P open parentheses 1.5 less or equal than X less than 5 close parentheses.

2d
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3 marks

Find straight P open parentheses X equals 4 vertical line X greater than 0 close parentheses.

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3a
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2 marks

Blue, a dog, barks an average of 25 times every hour.  His owner, Hayley, uses a Poisson distribution to model the number of times that Blue barks for any interval of time.

Write down two assumptions Hayley has made about Blue’s barks in order to use a Poisson distribution.

3b
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7 marks

Find the probability that Blue barks:

(i)
exactly 5 times in a 10-minute period,
(ii)
at most 4 times in a 15-minute period,
(iii)
more than 47 times but no more than 51 times in a two-hour period.

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4a
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2 marks

Grace, a grumpy toddler, attends nursery five days a week. The number of tantrums that Grace has in a day follows a Poisson distribution with variance 3.14.

(i)
Write down the distribution for the total number of tantrums that Grace has during a week at nursery. State any assumptions that are needed. 
(ii)
Hence, find the probability that Grace has exactly 17 tantrums during a week at nursery.
4b
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3 marks

Find the probability that Grace has fewer than four tantrums in a two-day period at nursery.

4c
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4 marks

Given that Grace has fewer than four tantrums at nursery one day, find the probability that she had no tantrums at nursery that day.

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5a
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4 marks

The table below shows the number of detentions per school day that Ms Ottway, a teacher, issues to students over a period of 150 days. 

 

Number of detentions

Number of days

0

51

1

54

2

36

3

6

4

3

5 or more

0

By calculating unbiased estimates for the mean and the variance, show that a Poisson distribution is an appropriate model for the number of detentions Ms Ottway issues.

 

5b
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3 marks

Using a Poisson distribution with the unbiased estimate of the mean, find the probability that Ms Ottway issues at least 5 detentions in a day.

5c
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4 marks

Students at Ms Ottway’s school attend school 5 days a week for 40 weeks a year.

Estimate the number of weeks in a school year that Ms Ottway issues fewer than 3 detentions.

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6
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7 marks

X comma space Y, and Z are random variables with W tilde Po open parentheses 75 close parentheses comma space X tilde Po open parentheses x close parentheses and Y tilde P o open parentheses y close parentheses. 

(i)
Find the largest integer value of a such that P open parentheses W less or equal than a close parentheses less than 0.5. 
(ii)
Given that straight P open parentheses X equals 0 close parentheses equals 0.253, find the value of x.
(iii)
Given that 5 space Var open parentheses Y close parentheses equals open parentheses straight E open parentheses Y close parentheses close parentheses squared minus 6, find the value of y.

 

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7a
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2 marks

In a game, enemies appear independently and randomly at an average rate of 2.5 enemies every minute. 

Find the probability that exactly 10 enemies will appear in a five-minute period.

7b
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3 marks

Find the probability that at least 3 enemies will appear in a 90-second period.

7c
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3 marks

The probability that at least one enemy appears in k seconds is 0.999. Find the value of k correct to 3 significant figures.

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8a
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3 marks

Matt has three best friends: Danny, Jessica and Luke.  Matt receives messages from Danny at an average rate of 1.7 messages per hour, from Jessica at an average rate of 1.5 messages per hour, and from Luke at an average rate of 1.1 messages per hour.

Stating any necessary assumptions, find the probability that Matt receives fewer than 4 messages from Danny in a two-hour period.

8b
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2 marks

Stating any additional assumptions needed, find the probability that Matt receives exactly 6 messages in total within a one-hour period.

8c
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2 marks

Find the probability that Matt will receive at least one message from any of the three friends within a one-hour period.

8d
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3 marks

Find the probability that Matt will receive at least one message from any of the three friends within a one-hour period.

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9a
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3 marks

Mick runs a business printing designs onto t-shirts.  He has five identical machines which he uses at the same time. The number of times that a given machine malfunctions within any given time interval can be modelled using a Poisson distribution. Whether or not a given machine malfunctions is independent of whether or not the other machines malfunction. On average, each machine malfunctions at a rate of 0.8 times per hour.

Find the probability that the combined total number of times that the five machines malfunction in a 10-hour period is fewer than 30.

9b
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4 marks

Each day the five machines operate for 10 hours. At the end of each day, Mick asks an engineer to check a machine if it has malfunctioned more than 12 times throughout the day.

Find the probability that at the end of a random day Mick will ask the engineer to check at least two of the machines.

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1a
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3 marks

State which distribution – normal, binomial or Poisson – is likely to be appropriate for calculating the final value of each of the following probabilities.  In each case specify any assumptions that would need to be made, and any parameters of which you would need to know the value in order to carry out the calculation.

The probability that the next person to walk through the door of a shop has a height of 1.7 metres or more.

1b
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3 marks

The probability that exactly 3 people with a height of 1.7 metres or more walk through the door of a shop in the next 20 minutes.

1c
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3 marks

The probability that of the next 12 people to walk through the door of a shop, exactly three of them have a height of 1.7 metres or more.

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2a
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2 marks

Amira has a bad Internet connection at her house.  Her internet disconnects on average 5 times each day.

Define a suitable distribution to model the number of times the internet at Amira’s house disconnects during a day. State any assumptions you make.

2b
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4 marks

Find the probability that during a random day the internet at Amira’s house disconnects:

(i)
exactly four times
(ii)
at most three times
(iii)
no fewer than two times.

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3a
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2 marks

Lucy loves the cinema and goes on average four times a week.  The number of times she goes to the cinema in a week can be modelled as a Poisson distribution with a mean of four times.

Find the probability that Lucy goes to the cinema exactly five times in a week.

3b
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2 marks

Find the probability that Lucy goes to the cinema no more than four times in a fortnight.

3c
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3 marks

Find the probability that Lucy goes to the cinema at least once in a day.

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4a
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2 marks

Comic Stans is a comic book store in the city of Krakoa.  Customers enter the store randomly and independently at an average rate of 8 people every 15 minutes.

Find the probability that exactly three people enter the store in a 1-minute period.

4b
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3 marks

Find the probability that someone enters the store in a 15-second period.

4c
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3 marks

Find the probability that at most three people enter the store in a 10-minute period.

4d
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2 marks

Find the variance of the number of people entering the store in a 1-hour period.

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5a
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2 marks

Amber suggests that she can model the number of times that she hiccups using a Poisson distribution.

Write down two conditions that must apply for this model to be applicable.

5b
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6 marks

The mean number of hiccups in a 30-second period is 2.9.

(i)
Assuming a Poisson distribution is applicable, find the probability that
(ii)
Amber hiccups exactly three times in a 30-second period
(iii)
Amber hiccups at least twice but no more than five times in a 15-second period
(iv)
Amber hiccups during a one-minute period.

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6a
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5 marks

The table below shows the data from a sample of 50 observations of a variable x.

(i)
Calculate unbiased estimates for the mean and the variance.
(ii)
State, with a reason, whether a Poisson distribution could be used to model the population’s data.

 x  0 1 3

Frequency

 3  5  17  15  10

6b
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5 marks

The table below shows the data from a sample of 100 observations of a variable y.

(i)
Calculate unbiased estimates for the mean and the variance.
(ii)
State, with a reason, whether a Poisson distribution could be used to model the population’s data.
 y  0 1

Frequency

 15  19  25  21  12  8

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7a
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4 marks

Jim is a bird watcher and is trying to model the number of birds that fly past his window.  During a 10-minute period he records the number of birds that fly past his window, and he repeats this a total of 120 times to form a sample.

Number of birds

Frequency

0

43

1

44

2

22

3

8

4

3

5 or more

0

Calculate unbiased estimates for the mean and the variance for the number of birds that fly past Jim’s window in a ten-minute period.

7b
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1 mark

Explain why a Poisson distribution would be appropriate to model the number of birds that fly past Jim’s window in a 10-minute period.

7c
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7 marks

Jim uses the distribution Po open parentheses 1 close parentheses to model the number of birds that fly past his window in a 10-minute period.

Use Jim’s model to calculate the probability that:

(i)
exactly two birds fly past Jim’s window in a 30-minute period
(ii)
fewer than two birds fly past Jim’s window in a 1-minute period
(iii)
at least four birds fly past Jim’s window in a 1-hour period.

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8a
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1 mark

Roberto orders a pizza from Pizza Palace and asks for two types of meat toppings: ham and salami.  It is known that the number of pieces of ham that Pizza Palace put on their pizzas follows a Poisson distribution with a mean of 6.2 pieces per pizza.  It is also known that the number of pieces of salami on their pizzas follows a Poisson distribution with a mean of 4.9 pieces per pizza.  The ham and salami are put on the pizza independently.

Write down the distribution that can be used to model the total number of pieces of meat on Roberto’s pizza.

8b
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2 marks

Find the probability that Roberto’s pizza contains a total of exactly 10 pieces of meat.

8c
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3 marks

Find the probability that Roberto’s pizza contains more than 9 but fewer than 13 pieces of meat.

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9a
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1 mark

André has is a keen amateur astronomer who spends his nights with a telescope trying to discover new comets.  Based on his past record of success, the number of times per year that André makes a new discovery may be modelled as a Poisson distribution with mean .

Use the model to find the probability that André makes exactly one new discovery in any given year.

9b
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5 marks

Over the course of three consecutive years, find the probability that André

(i)
makes exactly two new discoveries
(ii)
makes new discoveries in the second and third years only
9c
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5 marks

André’s partner Boglárka is a keen amateur entomologist who spends her spare time trying to discover new species of insects.  Based on her past record of success, the number of times per year that Boglárka makes a new discovery may be modelled as a Poisson distribution with mean 1.3.

Find the probability that, between them, André and Boglárka make at least one new discovery over a 3-month period, specifying any assumptions you make

9d
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4 marks

Find the probability that over a period of 12 years there will be exactly 4 years during which neither André nor Boglárka make a new discovery.

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1a
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2 marks

Determine, with a reason, whether or not the variable X in each of the following scenarios can be modelled by a Poisson distribution, Po open parentheses m close parentheses.  In each case where a Poisson distribution could be appropriate, specify any assumptions that would need to be made and identify the value for m that could be used.

Harry is walking along a beach in a straight line and finds on average 5 shells every 100 metres. X  is the distance Harry walks after finding a shell until he finds another shell.

1b
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2 marks

Each week Hermione donates £5 to a children’s charity. X is the amount of money that Hermione donates to the charity within a 10-week period.

1c
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2 marks

On average, Ron receives 7.2 emails every 45 minutes and Ginny receives 4.1 emails every 30 minutes. X is the combined number of emails that Ron and Ginny receive within a 60-minute period.

1d
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2 marks

Neville works an 8-hour shift at a call centre. Neville receives calls from customers at an average rate of 12.4 calls per hour. X is the number of calls that Neville receives during the last hour of his shift.

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2a
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2 marks

Phoebe has a faulty smoke detector which rings at an average rate of 5 times a day. 

Write down two conditions for a Poisson distribution to be a suitable model for the number of times that Phoebe’s smoke detector rings.

2b
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4 marks
(i) 

      Find the probability that Phoebe’s smoke detector rings exactly 3 times everyday over a four-day period.

(ii)      Find the probability that Phoebe’s smoke detector rings exactly 12 times in a four-day period.

(iii)     Explain why the answers to part (b)(i) and part (b)(ii) are different.
2c
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4 marks

Given that Phoebe’s smoke detector rang at least once in a 6-hour period, find the probability that Phoebe’s smoke detector rang no more than 4 times during that period.

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3a
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7 marks

The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled as a Poisson distribution with a mean of 1.2.  The number of mistakes made by a different student, Qays, in a 30-minute revision period is modelled as a Poisson distribution with a mean of 2.2.

Find the probability that Priya makes exactly 2 mistakes and Qays makes exactly 1 mistake within a one-hour revision period. State any assumptions that are needed.

3b
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3 marks

The number of mistakes made by Priya and Qays in a one-hour revision period are added together. Given that they make exactly 9 mistakes in total in a one-hour revision period, find the probability that Priya made exactly 5 mistakes in that same revision period.

3c
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2 marks

Given that Priya makes exactly 5 mistakes in a one-hour revision period, find the probability that Priya and Qays made exactly 9 mistakes in total in that same revision period.

3d
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2 marks

In a week before a test, Priya has ten 30-minute revision periods. Estimate the number of these revision periods during which Priya will make a mistake.

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4a
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3 marks

Whilst writing an essay, Gamu notices that she makes spelling mistakes at a rate of 7 for every 150 words.  Gamu models the number of spelling mistakes she makes using a Poisson distribution.

Find the maximum number of words Gamu can write before the probability of her making a spelling mistake exceeds 0.75.

4b
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6 marks

Gamu is asked to write three short essays by her lecturer.  She writes one containing 100 words, one containing 200 words and one containing 250 words. An essay is returned by Gamu’s lecturer if more than 1% of its words contain spelling mistakes.

Find the probability that

(i)
all three short essays are returned.
 
(ii)
exactly one of the three short essays is returned.

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5a
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6 marks

Caroline is observing cars as they stop at a stop sign outside of her shop.  She models the number of cars that stop outside her shop in a given time period using a Poisson distribution.  Caroline calculates that cars stop outside her shop at an average rate of 25.4 cars per hour.  

Caroline observes how many cars stop at the stop sign outside her shop each hour within a four-hour period.

Find the probability that:

(i)
fewer than 100 cars stop outside the shop within the four hours.
(ii)
at least 20 cars stop outside the shop within each hour of the four-hour period.
(iii)
more than 90 cars stop outside her shop within the four hours given that exactly 35 cars stop outside her shop within the first two hours.

 

5b
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4 marks

The number of people that walk past Caroline’s shop in an hour can be modelled as a Poisson distribution with mean 37.9. Caroline hands out a flyer advertising her shop to each person that walks past and to the driver of each car that stops outside her shop.

Caroline starts the day with 300 flyers. Find the probability that Caroline has no flyers left after five hours.  State an assumption that is needed.

5c
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2 marks

Caroline decides to hand out two flyers to the driver of each car that stops outside her shop and only one to each person that walks past. Explain, with a reason, whether or not a Poisson distribution could be used to model the total number of flyers that Caroline hands out within an hour.

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6a
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7 marks

Whilst playing the video game Thirsty Turkeys, a player receives points for various achievements.  Each week, Jamie plays the game for one hour each day from Monday to Friday.  From historical data, Jamie has found that the number of points he gets in an hour can be modelled by a Poisson distribution with mean 21. The number of points that Jamie gets on any one day is independent of the number of points he got on previous days.

If Jamie gets more than 25 points in a day then he gets a bronze star award.  If Jamie gets a bronze star award on at least three out of the five days in a week he gets a silver star award.  If Jamie gets a bronze star award on at least three consecutive days in a week then he gets a gold star award.

Find the probability that:

(i)
Jamie gets a silver star award in a given week
(ii)
Jamie gets a gold star award in a given week
(iii)
Jamie gets a gold star award in a given week, given that he gets a silver star award that same week.
6b
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4 marks

If Jamie gets at least 125 points in a five-day week then his name will appear on the leaders’ board.

Given that Jamie’s name appears on the leaders’ board one week, find the probability that he does not get any bronze star awards that week.

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7a
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3 marks

In 1898, statistician Ladislaus Bortkiewicz made one of the first applications of the Poisson distribution to a real-life situation. He observed 10 Prussian army units over a 20-year period and recorded how many soldiers died each year in each unit by accidentally being kicked by a horse.  His data is shown below: 

Number of accidental deaths by horse kick reported in a unit in a year

Frequency

0

109

1

65

2

22

3

3

4

1

5 or more

0

 

Use the data above to calculate unbiased estimates for the mean and variance of the number of accidental deaths by horse-kick reported in a unit in a year. Explain with a reason whether or not this supports Bortkiewicz’s suspicion that the data follows a Poisson distribution.

7b
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5 marks

Roger wants to make a fictional cartoon about unicorns accidentally poking their riders with their horns. He decides to allow the number of times per episode that characters accidentally get poked by a unicorn horn to be determined randomly, using a Poisson distribution with the mean from part (a). Roger plans to make three episodes. The Unicorn Protection Agency warn Roger that they will file a complaint if there is more than one instance of a character being poked by a unicorn’s horn in any one episode, or if there are more than two such instances in total throughout the three episodes.

Find the probability that the Unicorn Protection Agency will file a complaint.

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