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

Exam Questions

4 hours31 questions
1a
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2 marks

A random variable X space tilde space P o left parenthesis 5 right parenthesis.

Write down the integer values of a and b which make the following statements true:

(i)
P left parenthesis X less or equal than 7 right parenthesis equals 1 minus P left parenthesis X greater or equal than a right parenthesis
(ii)
P left parenthesis X greater or equal than 3 right parenthesis equals 1 minus P left parenthesis X less or equal than b right parenthesis.
1b
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6 marks

Use calculations of the form p subscript r equals e to the power of negative lambda end exponent fraction numerator lambda to the power of r over denominator r factorial end fraction with lambda equals 5 to find:

(i)
P left parenthesis X equals 4 right parenthesis
(ii)
P left parenthesis X less or equal than 2 right parenthesis
(iii)
P left parenthesis X greater or equal than 2 right parenthesis.


Give your answers to three significant figures.

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2a
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2 marks

A random variable X space tilde space P o left parenthesis 3 right parenthesis.

(i)
Write down the name of this distribution.
(ii)
Write down the mean number of occurrences, lambda.
2b
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4 marks

Use calculations of the form p subscript r equals e to the power of negative lambda end exponent fraction numerator lambda to the power of r over denominator r factorial end fraction to find exact values, in terms of e, for

(i)
P left parenthesis X equals 5 right parenthesis
(ii)
P left parenthesis X less or equal than 1 right parenthesis
(iii)
P left parenthesis X greater or equal than 2 right parenthesis.

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3a
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3 marks

The random variablespace X space tilde space P o left parenthesis lambda right parenthesis.

Find the value of space P left parenthesis X equals 0 right parenthesis in the case when:

(i)
lambda equals 7
(ii)
lambda equals 2.5
(iii)
lambda equals 0.01
3b
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3 marks

Find the value ofspace P left parenthesis X equals 1 right parenthesis in the case when:

(i)
lambda equals 7
(ii)
lambda equals 2.5
(iii)
lambda equals 0.01
3c
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2 marks

Given that space lambda equals k, find a simplified expression in terms of k for:

(i)
P left parenthesis X equals 0 right parenthesis
(ii)
P left parenthesis X equals 1 right parenthesis

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4
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3 marks

For each of the following random variables, calculate the mean using space mu equals lambda spaceand the standard deviation using space sigma equals square root of lambda.

(i)
X space tilde space P o left parenthesis 4 right parenthesis
(ii)
X space tilde space P o left parenthesis 1 right parenthesis
(iii)
X space tilde space P o left parenthesis 5.4 right parenthesis

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5a
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4 marks

The random variable space X space tilde space P o left parenthesis 6.3 right parenthesis.

(i)
Factorise the expression e to the power of negative lambda end exponent fraction numerator lambda to the power of 4 over denominator 4 factorial end fraction plus e to the power of negative lambda end exponent fraction numerator lambda to the power of 5 over denominator 5 factorial end fraction plus e to the power of negative lambda end exponent fraction numerator lambda to the power of 6 over denominator 6 factorial end fraction plus e to the power of negative lambda end exponent fraction numerator lambda to the power of 7 over denominator 7 factorial end fraction plus e to the power of negative lambda end exponent fraction numerator lambda to the power of 8 over denominator 8 factorial end fraction.
(ii)
Hence calculate space P left parenthesis 4 less or equal than X less or equal than 8 right parenthesis.
5b
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4 marks

Calculate:

(i)
P left parenthesis X less or equal than 3 right parenthesis
(ii)
P left parenthesis 5 less or equal than X less or equal than 7 right parenthesis

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6a
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4 marks

A random variable space X space tilde space P o left parenthesis 6 right parenthesis.

Find the exact value, in terms of straight e, of:

(i)
P left parenthesis X equals 3 right parenthesis
(ii)
P left parenthesis X equals 4 right parenthesis
(iii)
P left parenthesis X equals 5 right parenthesis
6b
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2 marks

Use your answers to part (a) to find the exact value of:

(i)
6 over 4 space P left parenthesis X equals 3 right parenthesis

(ii)
6 over 5 space P left parenthesis X equals 4 right parenthesis
6c
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1 mark

Compare your answers to part (a) and part (b), then use the pattern to write down an expression, in terms of k, for

fraction numerator 6 over denominator k plus 1 end fraction space P left parenthesis X equals k right parenthesis

where k is an integer and k greater or equal than 0.

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7a
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2 marks

Naomi, a teacher, receives emails at a constant average rate during the school day. Naomi models the number of emails she receives within a fixed period of time using a Poisson distribution, space P o left parenthesis lambda right parenthesis.

Write down two assumptions that Naomi has made about the emails in order to use a Poisson model.

7b
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3 marks

Naomi knows that on average she receives 10 emails in one hour.

Calculate the value of lambda in the case when Naomi is modelling the number of emails she receives in:

(i)
two hours
(ii)
15 minutes
(iii)
one hour and 10 minutes.

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8a
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4 marks

Use the tables of values for the Poisson cumulative distribution function to calculate the probabilities in this question.

The random variable X space tilde Po left parenthesis 6.5 right parenthesis, find:

(i)
P left parenthesis X less or equal than 10 right parenthesis
(ii)
P left parenthesis X greater than 7 right parenthesis
(iii)
P left parenthesis X equals 5 right parenthesis
8b
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4 marks

The random variable Y space tilde P o left parenthesis 4 right parenthesis, find:

(i)
P left parenthesis Y equals 0 right parenthesis
(ii)
P left parenthesis Y less than 7 right parenthesis
(iii)
P left parenthesis Y greater or equal than 5 right parenthesis.

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1a
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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.

1b
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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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2a
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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.

2b
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2 marks

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

2c
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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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3a
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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.

3b
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3 marks

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

3c
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3 marks

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

3d
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1 mark

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

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4a
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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.

4b
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7 marks

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.

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5a
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5 marks

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.

bold italic x 0 1 2 3 4
Frequency 3 5 17 15 10
5b
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5 marks

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.

bold italic y 0 1 2 3 4 5
Frequency 15 19 25 21 12 8

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6a
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3 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 estimates for the mean and the variance for the number of birds that fly past Jim’s window in a ten-minute period.

6b
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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.

6c
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6 marks

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.

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7a
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4 marks

The random variable X space tilde P o left parenthesis 10 right parenthesis.

Find:

(i)
P left parenthesis X less than 8 right parenthesis
(ii)
P left parenthesis X greater or equal than 11 right parenthesis
(iii)
P left parenthesis 5 less or equal than X less or equal than 14 right parenthesis.
7b
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2 marks

Find the largest integer k such P left parenthesis X greater than k right parenthesis greater than 0.25.

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1a
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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 left parenthesis X right parenthesis.

1b
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2 marks

Find space P left parenthesis X equals 4 right parenthesis.

1c
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3 marks

Find space P left parenthesis 1.5 less or equal than X less than 5 right parenthesis.

1d
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3 marks

Find space P left parenthesis X equals 4 space │ space X greater than 0 right parenthesis.

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2a
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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.

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

2b
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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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3a
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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.

Find the probability that Grace has exactly 17 tantrums during a week at nursery.

3b
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3 marks

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

3c
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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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4a
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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 estimates for the mean and the variance, show that a Poisson distribution is an appropriate model for the number of detentions Ms Ottway issues.

4b
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3 marks

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

4c
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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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5
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7 marks

X comma space Y and Z are random variables with X space tilde space space Po left parenthesis alpha right parenthesis comma space space Y space tilde space space Po left parenthesis beta right parenthesis and Z space tilde space space Po left parenthesis gamma right parenthesis.

(i)
Given that straight P left parenthesis X equals 3 right parenthesis equals straight P left parenthesis X equals 7 right parenthesis, find the value of alpha.
(ii)
Given that space P left parenthesis Y equals 0 right parenthesis equals 0.253, find the value of beta.
(iii)
Given that 5 Var left parenthesis Z right parenthesis equals left parenthesis E left parenthesis Z right parenthesis right parenthesis squared minus 6, find the value of gamma.

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6a
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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.

6b
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3 marks

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

6c
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3 marks

The probability that at least one enemy appears in k minutes is 0.999.  Find the value of k.

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7
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6 marks

The random variable X space tilde P o left parenthesis 10 right parenthesis. Find

(i)
the largest integer value of q such that P left parenthesis X less than q right parenthesis less than 0.2
(ii)
the largest integer value of r such that P left parenthesis X greater or equal than r right parenthesis greater than 0.4
(iii)
the smallest integer value of s such that P left parenthesis X greater than s right parenthesis less than 0.001.

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8a
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1 mark

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.

8b
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5 marks

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.
8c
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2 marks

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.

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1a
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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.

1b
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5 marks
(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.
1c
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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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2a
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3 marks

The random variable space X space tilde space space Po left parenthesis 9.5 right parenthesis.

(i)
Given that k is a non-negative integer, find the largest value of k such that space straight P left parenthesis X equals k right parenthesis less than straight P left parenthesis X equals k plus 1 right parenthesis .
(ii)
Hence find the mode of X.
2b
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4 marks

The random variable Y space tilde Po left parenthesis straight m right parenthesis  where m is a positive integer.

(i)
Show that P left parenthesis Y equals m right parenthesis equals P left parenthesis Y equals m minus 1 right parenthesis.
(ii)
Hence show that Y has two modes. State the two modes.

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3
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7 marks

The random variables X space tilde space space Po left parenthesis alpha right parenthesis comma space space Y tilde Po left parenthesis beta right parenthesis and Z tilde P o left parenthesis gamma right parenthesis.

(i)
Given that P left parenthesis X equals 4 right parenthesis equals P left parenthesis X equals 2 right parenthesis plus P left parenthesis X equals 3 right parenthesis find the value of alpha.
(ii)
Given that P left parenthesis Y greater or equal than 1 right parenthesis equals 0.314, find the value of beta.
(iii)
Given that P left parenthesis Z equals k plus 1 right parenthesis equals P left parenthesis Z equals k minus 1 right parenthesis where k is a positive integer, find an expression for gamma in terms of k.

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4
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6 marks

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:

Number of times hit by a flying bomb

Number of squares

0

 229

1

 211

2

 93

3

 35

4

 7

5 or more

 1

Roger suggests that a Poisson distribution with mean of 537 over 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.

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5a
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3 marks

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 a over b e to the power of c where a comma space b and c are integers to be found.  State any assumptions that are needed.

5b
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4 marks

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 q over r e to the power of s where q comma space r and s are integers to be found.

5c
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1 mark

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.

5d
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2 marks

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.

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6a
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4 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.

6b
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7 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 all three short essays are returned.

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7
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5 marks

Use the fact that Po left parenthesis lambda right parenthesis is a valid probability distribution, along with the properties of the Poisson distribution, to demonstrate the validity of the Taylor series expansion

e to the power of x equals 1 plus x plus fraction numerator x squared over denominator 2 factorial end fraction plus fraction numerator x cubed over denominator 3 factorial end fraction plus fraction numerator x to the power of 4 over denominator 4 factorial end fraction plus midline horizontal ellipsis

for all positive values of x.

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8a
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1 mark

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.

8b
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5 marks

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.
8c
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3 marks

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.

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