Poisson Distribution (Edexcel International A Level (IAL) Maths: Statistics 2): Exam Questions

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.

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.

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.

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

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

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.

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

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

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

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.

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

Assuming a Poisson distribution is applicable, find the probability that

(i) Amber hiccups exactly three times in a 30-second period

(ii) Amber hiccups at least twice but no more than five times in a 15-second period

(iii) Amber hiccups during a one-minute period.

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

(i) Calculate 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.

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

(i) Calculate 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.

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

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.

Jim uses the distribution  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.

The random variable $X~Po\left(10\right)$.

Find:

(i) $P(X<8)$

(ii) $P(X\ge 11)$

(iii) $P(5\le X\le 14)$.

Find the largest integer $k$ such $P(X>k)>0.25$.

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

Write down the value for $\mathrm{E}\left(X\right).$

Find $P(X=4)$.

Find $P(1.5\le X<5)$.

Find $P(X=4│X>0)$.

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.

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

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.

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 exactly 17 tantrums during a week at nursery.

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 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 estimated 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.

$X,Y$ and $Z$ are random variables with $X~\mathrm{Po}\left(\alpha \right),Y~\mathrm{Po}\left(\beta \right)$ and $Z~\mathrm{Po}\left(\gamma \right)$.

(i) Given that $\mathrm{P}(X=3)=\mathrm{P}(X=7)$, find the value of $\alpha$.

(ii) Given that $P(Y=0)=0.253$, find the value of $\beta$.

(iii) Given that $5\mathrm{Var}\left(Z\right)=\left(E\right(Z){)}^2-6$, find the value of $\gamma$.

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 $k$ minutes is 0.999.  Find the value of $k$.

The random variable $X~Po\left(10\right)$. Find

(i) the largest integer value of $q$ such that $P(X<q)<0.2$

(ii) the largest integer value of $r$ such that $P(X\ge r)>0.4$

(iii) the smallest integer value of $s$ such that $P(X>s)<0.001$.

Matt receives text messages from his friends at an average rate of 4 messages per half hour.

State the assumptions needed to use a Poisson distribution to model the number of text messages Matt receives in a fixed time period.

Find the probability that

(i) Matt receives less than 5 text messages in a 30-minute period,

(ii) Matt receives no fewer than 3 text messages in a 15-minute period,

(iii) Matt receives at least one text message in a 20-minute period.

Matt’s friend, Jessica, bets that Matt will not receive more than $n$ text messages in the next hour.  Jessica wants the probability of her losing the bet to be less than 5%.

Find the smallest value of $n$ that Jessica should use.

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.

(i) Find the probability that Phoebe’s smoke detector rings exactly 3 times every day 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.

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 random variable $X~\mathrm{Po}(9.5).$

(i) Given that $k$ is a non-negative integer, find the largest value of $k$ such that $\mathrm{P}(X=k)<\mathrm{P}(X=k+1)$ .

(ii) Hence find the mode of $X$.

The random variable $Y~\mathrm{Po}\left(\mathrm{m}\right)$  where $m$ is a positive integer.

(i) Show that $P(Y=m)=P(Y=m-1)$.

(ii) Hence show that $Y$ has two modes. State the two modes.

The random variables $X~\mathrm{Po}\left(\alpha \right),Y~\mathrm{Po}\left(\beta \right)$ and $Z~Po\left(\gamma \right)$.

(i) Given that $P(X=4)=P(X=2)+P(X=3)$ find the value of $\alpha$.

(ii) Given that $P(Y\ge 1)=0.314$, find the value of $\beta$.

(iii) Given that $P(Z=k+1)=P(Z=k-1)$ where $k$ is a positive integer, find an expression for $\gamma$ in terms of $k$.

Roger is investigating historical data from World War II.  In south London, an area of 144 km² was divided into 576 equal squares and it was recorded how many times each square was hit by a flying bomb over a period of time.  During this time, it was recorded that a total of  bombs landed in the area.  The results are as shown in the table below:

Roger suggests that a Poisson distribution with mean of $\frac{537}{576}$ would be an appropriate model for the number of times a square was hit by a flying bomb.

(i) By finding probabilities using Roger’s model, estimate the expected number of squares that would hit, respectively, by 0, 1, 2, 3, 4 and 5 or more flying bombs.

(ii) By comparing the actual number of squares in the table with the expected number of squares from part (a)(i), state whether Roger’s model is appropriate.

The number of mistakes made by a student, Priya, in a 20-minute revision period is modelled as a Poisson distribution with 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. Write your answer in the form $\frac{a}{b}{e}^c$ where $a,b$ and $c$ are integers to be found.  State any assumptions that are needed.

Priya and Qays add together the number of mistakes they make after each revision session. Find the probability that they make a total of exactly 3 mistakes within a one-hour revision period. Write your answer in the form $\frac{q}{r}{e}^s$ where $q,r$ and $s$ are integers to be found.

Given that Priya makes exactly 2 mistakes in a one-hour revision period, find the probability that Priya and Qays made exactly 3 mistakes in total in that revision period.

Given that Priya and Qays make exactly 3 mistakes in total in a one-hour revision period, find the probability that Priya made exactly 2 mistakes in that revision period.

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 all three short essays are returned.

Use the fact that $\mathrm{Po}\left(\lambda \right)$ is a valid probability distribution, along with the properties of the Poisson distribution, to demonstrate the validity of the Taylor series expansion

$e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots$

for all positive values of $x$.

Jenna manages a team of extraordinary scientists.  The scientists work six days a week and make new discoveries at an average rate of one per week.

State any assumptions needed in order to use a Poisson distribution to model the number of new discoveries made by the scientists in a fixed time period.

Find the probability that

(i) the scientists make at least 8 new discoveries within a five-week period,

(ii) the scientists make no more than 3 new discoveries in a two-week period,

(iii) the scientists make at least one new discovery in one working day.

Jenna is planning a new project on which the scientists will work for $k$ weeks, where $k$ is an integer.  Jenna wants there to be a probability of at least 1% that the number of new discoveries made by the scientists is more than double the number of weeks of the project.

Find the largest value of $k$ that Jenna should choose.