nuExam07 - Binomial Distribution - Numbas at mathcentre.ac.uk

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Michael Proudman 5 years, 10 months ago

Created this as a copy of Exam07 - Binomial Distribution.

Name	Status	Author	Last Modified
Clodagh's copy of Poisson Distribution (printing errors)	draft	Clodagh Carroll	05/03/2021 12:07
Binomial Distribution (Cycling)	Ready to use	Julie Crowley	26/10/2023 12:21
Julie's copy of Clodagh's copy of Poisson Distribution (printing errors)	draft	Julie Crowley	07/03/2017 15:48
STAT7008 (Cycling)	draft	Catherine Palmer	06/10/2018 22:00
STAT7008 (printing errors)	Ready to use	Catherine Palmer	07/12/2021 15:12
Simon's copy of Binomial Distribution (Cycling)	draft	Simon Thomas	28/03/2019 12:17
Simon's copy of Clodagh's copy of Poisson Distribution (printing errors)	draft	Simon Thomas	28/03/2019 12:36
Simon's copy of Clodagh's copy of Poisson Distribution (printing errors)	draft	Simon Thomas	28/03/2019 14:36
Maria's copy of Binomial Distribution (Cycling)	draft	Maria Aneiros	27/05/2019 02:54
Exam07 - Binomial Distribution	draft	Michael Proudman	07/12/2021 09:54
nuExam07 - Binomial Distribution	draft	Michael Proudman	07/12/2021 09:54
Poisson Distribution (Dentist)	Ready to use	Catherine Palmer	22/03/2024 08:44
Binomial Distribution (Cycling)	Ready to use	Yvonne Wancke	26/06/2023 16:36
Binomial Distribution walks	Ready to use	Yvonne Wancke	26/06/2023 16:37
Poisson Distribution (Dentist)	Ready to use	Yvonne Wancke	26/06/2023 16:43
S2 EXAM Binomial Distribution 1	Ready to use	Yvonne Wancke	26/06/2023 16:34
S2 EXAM Poisson Distribution 1 doctors	Ready to use	Yvonne Wancke	26/06/2023 16:35

There are 73 other versions that do you not have access to.

Question statement

Please give your answer to at least 3 decimal places.

It is estimated that $\var{p_perc}$ % of all Lakes College students walk to college. A random sample of $\var{n}$ Lakes College students is chosen.

Give any introductory information the student needs.

			Name	Type	Generated Value

p	number	0.15
p_perc	number	15
n	integer	9
q	number	0.85
r	integer	3
pr0	number	0.2316169463
pr1	number	0.3678622088
pr2	number	0.2596674415
pr3	number	0.1069218877
answer1	number	0.8591465966
answer2	number	0.1408534034
qn	number	0.2316169463
r0	integer	0
n2	integer	7

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(0.1..0.2#0.05)

JME function reference

Description

the probability that an individual student cycles to college

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `number`

0.15

→ Used by:

p_perc
pr1
pr2
pr3
q

This variable doesn't seem to be used anywhere.

Advice

Part (a)

If a random variable $X$ follows a binomial distribution with parameters $n$ and $p$ . The probability of $r$ successes out of $n$ trials is given by:

$P(X=r)=P(r,n)=C^n_{r}p^{r}q^{n-r}$

where $p$ is the probability of success for each trial and $q$ is the probability of failure for each trial.

The probability that a student cycles to college is $\var{p}$ , therefore $p=\var{p}$ and $q=1-\var{p}=\var{q}$ .

We are interested in claculating the probability that none of the sample of $\var{n}$ students walk to college so $r=0$ and $n=\var{n}$

$P(\var{r0}, \var{n})= C^\var{n}_{\var{r0}}$ $\var{p}^\var{r0}$ $\var{q}^{\var{n}-\var{r0}}$

$P(\var{r0}, \var{n})= \var{pr0}$

Part (b)

We are interested in claculating the probability that at least $\var{r}$ of the $\var{n}$ students walk to college. Let $X$ represent the number of students that walk to college. We need to calculate:

$P(X \geq \var{r}) = P(X= \var{r}) + P(X= \var{r+1})+...+ P(X=\var{n})$

Since $P(X=\var{r0})+P(X=\var{r0+1})+...+P(X=\var{n})=\var{r0+1}$

We may write

$P(X \geq \var{r}) = 1-P(X= \var{r0}) - P(X=\var{r0+1})-...- P(X=\var{r-1})$

where

$P(X= \var{r0})=P(\var{r0}, \var{n})= C^\var{n}_{\var{r0}}$ $\var{p}^\var{r0}$ $\var{q}^{\var{n}-\var{r0}}=\var{pr0}$

$P(X=1) =P(1, \var{n})= C^\var{n}_{1}$ $\var{p}^{1}$ $\var{q}^{\var{n}-1}$ $=\var{pr1}$

$\var{p}^{2}$ $\var{q}^{\var{n}-2}$ $=\var{pr2}$

Then

$P(X \geq \var{r}) = 1-\var{qn}-\var{pr1}-\var{pr2}=\var{answer2}$

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This question is used in the following exams:

HNC Engineering Maths Exam by Michael Proudman in Assessment Exercises.
Probability 2 by Jeremie Wenger in Jeremie's workspace.
gills numas try by Gill McCartney in Gill's workspace.
Paul's copy of CMPU2012-Probability by Paul Molloy in Paul's workspace.
CMPU2012-Probability by Blathnaid Sheridan in Blathnaid's workspace.

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nuExam07 - Binomial Distribution

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Michael Proudman 5 years, 10 months ago

Error in variable testing condition

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nuExam07 - Binomial Distribution

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Michael Proudman 5 years, 10 months ago

Error in variable testing condition

Error

Warning

Generated value: number

Parts

Number entry

Advanced settings

Generated value: `number`