Probability of the union of two non-mutually exclusive events

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Christian Lawson-Perfect	said	Ready to use	7 years, 7 months ago
Elliott Fletcher	said	Needs to be tested	7 years, 8 months ago
Chris Graham	said	Has some problems	7 years, 8 months ago

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Christian Lawson-Perfect 7 years, 7 months ago

Gave some feedback: Ready to use

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Christian Lawson-Perfect 7 years, 7 months ago

Saved a checkpoint:

After being "fixed", the statement was incorrect: the first two items were for only eating cereal or toast, so the probability of eating either is the sum of all three items. This isn't what was intended.

After lots of back and forth, I think we've settled on a wording that works. I've changed the statement to give a table with five options, adding up to over 100%, to make it clear that the percentages include people who selected more than one option.

Restore this checkpoint

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Elliott Fletcher 7 years, 8 months ago

Published this.

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Elliott Fletcher commented 7 years, 8 months ago

Thanks for the feedback Chris, i have tried to reword the statement, hopefully it is clearer now.

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Chris Graham 7 years, 8 months ago

Gave some feedback: Has some problems

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Chris Graham commented 7 years, 8 months ago

I think that the statement needs re-wording. As it stands, to me, it suggests that the people having "cereal and toast" are in addition to the others. I can't quite put my finger on it, perhaps the presence of "and"? Try this on Aiden and see what he thinks.

In the advice, "The Venn diagram tells us..." is not required, as you have just created the Venn diagram from the info given.

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Elliott Fletcher 7 years, 8 months ago

Gave some feedback: Needs to be tested

○

Elliott Fletcher 7 years, 8 months ago

Created this.

Name	Status	Author	Last Modified
Probability of the union of two non-mutually exclusive events	Ready to use	Elliott Fletcher	20/11/2019 14:41
Simon's copy of Probability of the union of two non-mutually exclusive events	draft	Simon Thomas	08/05/2019 11:51
Probability of the union of two non-mutually exclusive events	draft	Xiaodan Leng	11/07/2019 02:04

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Question statement

A survey asked people what they eat for breakfast. Participants had to select foods that they typically eat for breakfast from a list.

A story in the newspaper displayed the results of the survey in this table:

Food	Cereal	Toast	Fruit	Fry-up	Other
% of participants	{proportions[0]}	{proportions[1]}	{proportions[2]}	{proportions[3]}	{proportions[4]}

The most popular combination was cereal and toast, with $\var{b}\%$ of the participants selecting both.

Give any introductory information the student needs.

			Name	Type	Generated Value

c	integer	40
t	integer	20
b	integer	6
raw_proportions	list	List of 5 items
proportions	list	[ 40, 20, 17, 25, 26 ]
total	integer	131

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
proportions[0]//random(20..40)

JME function reference

Description

Percentage of people who have cereal for breakfast.

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

Can an exam override the value of this variable?

Generated value: `integer`

← Depends on:

proportions

→ Used by:

This variable doesn't seem to be used anywhere.

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Ask the student a question, and give any hints about how they should answer this part.

Consider the event that someone has cereal for breakfast and the event that the same person has toast for breakfast. Are these events mutually exclusive?

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(not displayed in columns)

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Yes

Marks: Distractor Message:
No

Marks: Distractor Message:

Advice

a)

Mutually exclusive events are events that cannot happen at the same time.

We know from the results of the survey that $\var{b}\%$ of participants stated that they have cereal as well as toast for breakfast.

Therefore it is possible to have both cereal and toast for breakfast, which means that the events "cereal" and "toast" are not mutually exclusive.

b)

We know from the results of the survey that some people have both cereal and toast for breakfast, so we can present the information given to us in the question in the form of a Venn diagram.

The number of people who have cereal or toast for breakfast is:

all the people who have cereal (including the participants who have cereal as well as toast)
all the people who have toast (including the participants who have toast as well as cereal)

However, this counts the people who have cereal as well as toast twice!

To correct our answer, we subtract the extra "and" part:

As a general formula this is:

$\mathrm{P}(\mathrm{A} \cup \mathrm{B}) = \mathrm{P}(\mathrm{A}) + \mathrm{P}(\mathrm{B}) - \mathrm{P}(\mathrm{A} \cap \mathrm{B}).$

Note that here we have made use of some notation that is frequently used in probability calculations:

The "Intersection" symbol $\cap$ , used instead of "and".
The "Union" symbol $\cup$ , used instead of "or".

Using this equation, the probability that a participant selected at random will either have cereal or toast for breakfast is

$\begin{align} \mathrm{P}(\text{cereal} \cup \text{toast}) &= \mathrm{P}(\text{cereal})+\mathrm{P}(\text{toast}) - \mathrm{P}(\text{cereal} \cap \text{toast})\\ &= \var{{c}/100}+\var{{t}/100}-\var{{b}/100}\\ &= \var{({c}+{t}-{b})/100}. \end{align}$

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Probability of the union of two non-mutually exclusive events

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Taxonomy: mathcentre

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Taxonomy: Context

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Christian Lawson-Perfect 7 years, 7 months ago

Christian Lawson-Perfect 7 years, 7 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Error in variable testing condition

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Generated value: `integer`

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b)

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Probability of the union of two non-mutually exclusive events

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Christian Lawson-Perfect 7 years, 7 months ago

Christian Lawson-Perfect 7 years, 7 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Chris Graham 7 years, 8 months ago

Chris Graham commented 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Elliott Fletcher 7 years, 8 months ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Choose one from a list

Advanced settings

a)

b)

Generated value: `integer`