Probability with Venn Diagrams: Learn It 1

  • Understand the concept of set theory and how it relates to probability
  • Create and interpret Venn diagrams to visually represent sets and their intersections
  • Understand how to use Venn diagrams to solve problems related to probability, including union, intersection, and complement of events

Set

set

A set is collection of distinct objects, called elements of the set.

 
The elements of a set can be either numbers or objects that have a particular characteristic in common. When defining a set, it is common to use a capital letter to define the set accompanied by the list of elements.
 
For example, we can define the set [latex]A = \{...\}[/latex] where the elements of the set are listed inside the curly bracket.

A set can contain both a finite or infinite number of elements depending on how the set is defined. The set of real numbers, [latex]R[/latex] is example of a set commonly used in math that has an infinite number of elements. In probability, we will typically focus on sets that have a finite number of elements.

Write out the following set of elements for each of the following sets.

(a) The set of letters in the word “MATH”.


(b) The set of primary colors.


(c)The set of counting numbers from 1 to 10.


(d) The set of odd numbers from 1 to 6.

One of the operations we can perform with a set is to determine the number of elements in the collection. For instance, the set [latex]A = \{\text{red, blue, green}\}[/latex] contains three elements. With these concepts in mind, sets can be valuable when examined in the context of probability. 

The sample space of a chance experiment is the collection of all possible outcomes for the experiment.

When analyzing a chance experiment, both the sample space [latex]S[/latex] and the event [latex]A[/latex] consist of sets of outcomes. By understanding the number of elements of these sets, we can calculate the probability of event [latex]A[/latex] by dividing the number of elements of the sets with the number of elements in the sample space.

Theoretical probability is the probability that an event will happen based on pure mathematics, not by carrying out an experiment.

Notation: [latex]P(\text{event})[/latex] indicates “probability of an event”.

probability

When the outcomes of the sample space are equally likely, the probability of an event is the number of elements in the event divided by the number of elements in the sample space.

 

[latex]P(\text{event}) = \dfrac{\text{number of elements in event}}{\text{number of all possible elements}}[/latex]

Suppose that carnival wheel is numbered 1 through 9. The wheel is spun at random and a prize is awarded when the number lands on an even number. Let [latex]A =[/latex] “the number is even”.

(a) Write out the sample space [latex]S[/latex] using set notation.

(b) Write out the event [latex]A[/latex] using set notation.

(c) How many elements are in the set [latex]S[/latex]?

(d) How many elements are in the set [latex]A[/latex]?

(e) Using your results, what is the probability of the event [latex]A[/latex]?