Probability and Venn Diagrams€¦ · What is a Venn Diagram Venn Diagrams are illustrations used in different branches of mathematics such as Set Theory or Probability. They show

Probability and Venn Diagrams

Coach Whitt

Probability and Venn Diagrams

�What can we say about probability?– It is a measure of likelihood, uncertainty,

possibility, …

– And it is a number, numeric measure, between 0 and 1.

�Simply put:– It is the chance that something will happen.

�For every event we can carefully assign a number which lies between 0 and 1 inclusive.

Experimental vs Theoretical

Probability� Experimental Probability

– Determined by observing the results of an experiment.

• Experimental probability = Number of event occurrences

Total number of trials

� Theoretical Probability– The likelihood in an event of what SHOULD happen.

• Theoretical probability = Number of favorable outcomes

Total number of outcomes

� Conditional Probability

– the probability that event B occurs given that event A has already occurred.

Probability Terms

� Compound Events– made up of two or more simple events.

• The probability that a flight will be on time and the luggage arrives is an example of a compound event.

� Dependent Events– 2 or more events whose outcomes affect each other.

The probability of occurrence of one event depends on the occurrence of the other.

� Independent Events– 2 or more events whose outcomes do not affect each

other.

Probability Terms

� Mutually Exclusive Events– 2 outcomes or evens are mutually exclusive when they

cannot both occur simultaneously.

� Mutually Inclusive Events– 2 outcomes or events are mutually inclusive when they

can both occur simultaneously.

� Outcome– The possible results after a number trials in an

experiment.

What is a Venn Diagram

What is a Venn Diagram

� Venn Diagrams are illustrations used in different branches of mathematics such as Set Theory or Probability.

� They show the mathematical or logical relationship between different groups of things (sets).

� A Venn Diagram shows all the possible logical relations between the sets.

Contents

� Who is credited with discovering Venn Diagrams

� What does a Venn Diagram look like

� How to use a Venn Diagram

Who created Venn Diagrams� Venn Diagrams are named after

John Venn, a British philosopher and mathematician who introduced these in 1881.

� John Venn used the diagrams to illustrate ideas used in logic.

� A stained glass window in Caius College, Cambridge, where he studied and spent most of his life, commemorates John Venn and represents a Venn Diagram.

What does a Venn Diagram look

like?� The simplest and most common Venn Diagram is made of a

rectangle and two circles that have an overlapping region.• See picture belowTerms –

•Sample Space or Universal Set

(Rectangle)

• Contains all things being

considered

• Represented by the red

rectangle

•Sets (Circles)

• Different groups of things

being considered

• The intersecting part of the

circles contains numbers that

can be found in both sets

Venn Diagram Terms

� Sample Space– A sample space for an experiment is the set of

ALL experimental outcomes.

� Set: – A set is a collection of distinct objects,

considered as an object in its own right– Example:

• A class of AQR students• NBA teams• Result of a football game for a team (win, lose, tie)• Outcomes of rolling a die ( 1, 2, 3, 4, 5, 6)

Terms

� Union

– “A union B” is the set of all elements that are in A, or B, or both.

A B∪

Terms

A B

A B∩

•Intersection

•“A intersect B” is the set of all

elements that are in both A and B. A B∩

Terms

� Complement of an event A: AC

– Includes ALL sample points that are not in a set.

Mostly Terrible Movies

How to Use a Venn Diagram

� Venn Diagrams can be used to show relationships between sets of information

� Each circle represents a set of information

Turtles

Ninjas

Ninja Turtles

• There can be more than 2 circles in a Venn Diagram

• The more circles (Sets) in a Diagram,

the more complicated it can be.