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Minds on

MINDS ON

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This is an image of an ill young girl with fever drinking a cup of warm tea and half-sleeping in a blanket.

At the start of the flu season, a doctor examines 50 patients over two days. 30 have a headache, 24 have a cold, 12 have neither. Some patients have both symptoms.

  • What is the probability that a random patient has both symptoms?
  • How can you use what you know about theoretical probability to solve this problem?
  • What are the challenges with this problem?
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Action.

ACTION

Mutually and Non-Mutually Exclusive Events

In the previous exercise, you learned that you can figure out difficult probabilities by making a simulation and using the Law of Large Numbers.  Theoretical probabilities can be calculated using many different strategies depending on the situation.

This is an image of a man pushing some buttons on a car radio with his wife sitting in the passenger seat.
You can drive and listen to the radio at the same time.

Mutually exclusive events are events that can not happen at the same time. Examples include: right and left hand turns, even and odd numbers on a die, winning and losing a game, or running and walking.  

Non-mutually exclusive events are events that can happen at the same time. Examples include: driving and listening to the radio, even numbers and prime numbers on a die, losing a game and scoring, or running and sweating.

Non-mutually exclusive events can make calculating probability more complex.

 

This is the ePortfolio icon. Games Fair Reflection

Play the Single Card Flip game in the Games Fair again.  

Applying the definition of mutually exclusive events, are the different points outcomes in the game mutually exclusive or non-mutually exclusive?

Take some time to write down some thoughts about how to calculate the probabilities.

GamesFair

Long Description

 
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The problem with only understanding probability as mutually exclusive events is that you are simplifying these events in a way that they aren’t meant to be simplified. It’s almost as tragic as saying that people can’t do good and make money, or be good at mathematics and very creative. The following video demonstrates the importance of recognizing events as non-mutually exclusive.  

 

Venn Diagrams

This is a Venn diagram that displays all of the cards in a standard deck of 52 cards. A is the set of non-face cards.

This is an image of a venn diagram representing a standard deck of cards.  Inside a rectangle is one circle, representing non-face cards called A.  n of A is equal to 40 and P of A is equal to 40 over 52 which equals 10 over 13.  Outside the circle is the face cards called A apostrophe.  n of A apostrophe is 12 and P of A apostrophe is 12 over 32 which equals 3 over 13.  The rectangle is called S and n of S equals 52.
A is the set of non-face cards and A' is face cards. Only one circle is used because an element is either in the circle or outside it. All cards, S, are in the rectangle.

Venn diagrams are a way to display events and can be used to display simple situations when only one event occurs.

They can also be used to display multiple events.

They are particularly useful when displaying non-mutually exclusive events.

In this activity you will use them to display 2 or 3 events at a time to analyse the relationship between the events.

The Intersection of Sets

This is an image of a Venn Diagram, two black circles with an overlapping section coloured red.
The red represents the intersection of sets.

Consider the Venn diagram for the 50 patients question from Minds On.

This is an image of a venn diagram representing flu symptoms. Inside the rectangle are two ovals, one called A which represents Headache and one called B which represents Cold. The ovals overlap in the middle.

We use the notation A upside down U B as the “Intersection” of the two sets, an element in A and B. 

In this example, A upside down U B represents the overlap of the symptoms, or the patients that have both headache “AND” flu symptoms.

You will come to know the intersection as the word “AND.”  

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You have been asked to find out how many people are in the "AND" category in order to calculate the probability that a patient has both symptoms. Rest assured, there is only one way to do it.

Use the interactive below to create a venn diagram to find the number of people that have both symptoms in the example.

At the start of the flu season, a doctor examines 50 patients over two days. 30 have a headache, 24 have a cold, 12 have neither. Some patients have both symptoms. What is the probability that a random patient has both symptoms?

VennDiagram

Long Description

In Words

In Symbols

In Symbols

All red

 n(A) =

P(A) =

All number

n(B) =

P(B) =

All cards

n(S) =

P(S) =

Intersection: Both number and red

n of A upside down U B =

P of A upside down U B =

Only Red (red cards that are not number cards)

n of A minus n of A intersect B =

P of A minus P of A intersect B =

Only Number (number cards that are not red cards)

n of B minus n of A intersect B =

P of B minus P of A intersect B =

Union:  Red or Number

n of A U B =

P of A U B =

Everything else

n of A U B complement =

P of A U B complement =

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The Principle of Inclusion-Exclusion

Consider the Venn Diagram. If you add n(A) and n(B), you count the intersection part twice. You should never count an element twice to find out how many you have. If you have counted it twice, there is an easy way to correct for that: subtract it once.

The Principle of Inclusion-Exclusion is a useful formula/idea when determining probabilities and is used for non-mutually exclusive events.

It can be written as follows:

n of A U B equals n of A plus n of B minus n of A intersect B

or in words:  the number of elements in A “OR” B is equal to the number of elements in A plus the number of elements in B subtract the number of elements in A “AND” B. 

 

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Use the formula that represents the principle of inclusion-exclusion to solve the original problem:  

At the start of the flu season, a doctor examines 50 patients over two days. 30 have a headache, 24 have a cold, 12 have neither. Some patients have both symptoms. What is the probability that a random patient has both symptoms?

Show your steps in your solution and explain (definition:Take time to write out explanations for anything that is not obvious or that took some thinking that is otherwise not shown.) your answer.

Solution

  • 8 have just a cold,
  • 14 have just a headache,
  • 16 have both.
 
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Venn Diagram Using 3 Events

Venn diagrams and mutually exclusive events are not only used with two different events.  Three events, although more complicated, can also be used in a Venn Diagram. If the proper strategy is applied, this can be done in an effective way.

This is the example icon. Example

This is an image of students working in small groups and studying in a library, some are standing, most are sitting.

The Student Services Department at Eastside Secondary wants to count the number of students in Grade 12. They know that every student is taking Math, English or Science. They found that:

  • 64 students are taking Math
  • 56 students are taking English
  • 82 students are taking Science
  • 20 students are taking Math and English
  • 25 students are taking Math and Science
  • 21 students are taking English and Science
  • 12 students are taking all three courses.

Create a three event Venn diagram similar to the one below. Fill in each section of your Venn diagram paying careful attention that each student is in only one category. 

  1. How many total students are there?
  2. What is the probability of a random student taking Math and Science?
  3. What is the probability of a random student taking Math or Science?
  4. What is the probability that a student takes exactly 2 of the 3?
 

This is an image of a venn diagram. Three overlapping circles in a rectangle that create different sections where any of the 2 overlap, and all three overlap.

 

Solution

  • 12 all three,
  • 9 in E and S but not M,
  • 13 in M and S but not E,
  • 8 in M and E but not S,
  • 48 in just S,
  • 27 in just E,
  • 31 in just M.

For help with this question, refer to the following similar example.  

 
test text.