Poisson Distribution (DP IB Maths: AI HL)

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.

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

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

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

Write down the value for .

Find .

Find 

Find .

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.

Find the probability that Blue barks:

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.

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

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

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. 

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.

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.

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.

, and  are random variables with  and  

(i)

Find the largest integer value of such that  

(ii)

Given that , find the value of .

(iii)

Given that , find the value of .

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.

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

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

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.

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

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

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

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.

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.

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

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

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

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

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. is the number of calls that Neville receives during the last hour of his shift.

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.

      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.

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.

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.

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.

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.

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.

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.

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

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:

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.

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.

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:

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.

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: 

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.

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.