Simon's copy of Probability of the union of two non-mutually exclusive events

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Simon Thomas 5 years, 11 months ago

Created this as a copy of Probability of the union of two non-mutually exclusive events.

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	42
t	integer	21
b	integer	7
raw_proportions	list	List of 5 items
proportions	list	[ 42, 21, 26, 30, 31 ]
total	integer	153

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.

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Generated value: `integer`

← Depends on:

proportions

→ Used by:

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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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Advanced settings

Selection type

Number of display columns

(not displayed in columns)

Custom marking matrix?

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.

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

all the people who have cereal (including the participants who have cereal AND toast)
all the people who have toast (including the participants who have toast AND cereal)

If we added these two numbers, this would count the people who have cereal AND toast twice! To correct our answer, we must subtract the number of people who had cereal AND toast.

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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This question is used in the following exam:

Introduction to probability by Simon Thomas in Maths support.

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Simon's copy of Probability of the union of two non-mutually exclusive events

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Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

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Checkpoint description

Simon Thomas 5 years, 11 months ago

Error in variable testing condition

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Generated value: `integer`

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a)

b)

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Simon's copy of Probability of the union of two non-mutually exclusive events

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

History

Comment

Checkpoint description

Simon Thomas 5 years, 11 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`