Probability with Venn Diagrams: Learn It 4

  • 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

intersection of events

The intersection of two events, [latex]A[/latex] and [latex]B[/latex], is denoted by [latex]A \cap B[/latex].

 

[latex]A[/latex] and [latex]B[/latex] means that all elements in event [latex]A[/latex] also belong in event [latex]B[/latex] or equivalently, all elements of [latex]B[/latex] is also in event [latex]A[/latex].

 

In terms of probability, [latex]P(A \text{ and }B)= P(A \cap B)=[/latex] the relative frequency of event [latex]A[/latex] and [latex]B[/latex] with respect to the sample space.

Consider the set [latex]A = \{\text{red}, \text{blue}, \text{green}\}[/latex] and [latex]B=\{\text{red},\text{yellow},\text{orange}\}[/latex].
The event [latex]A \cap B = \{\text{red}\}[/latex].

Using Venn diagrams, we can imagine AND events as the region where two events overlap. This is represented by the colored region in the Venn diagram below where two circles representing events [latex]A[/latex] and [latex]B[/latex] intersect.

     This Venn diagram depicts two overlapping circles within a rectangle labeled "S". The left circle is labeled "A", the right circle is labeled "B", and the overlapping area between the two circles is labeled "A and B".

A survey of [latex]220[/latex] random participants was conducted studying the behavior of adults age 18 – 25 years old.  According to the study, [latex]123[/latex] of them were attendance college and [latex]153[/latex] of them were working. Furthermore, [latex]56[/latex] of them were both attending college and working.

(a) Create a Venn diagram using the information given where C = “the event the participant attends college” and W = “the event the participant is working”.


(b) What is the probability that the participant is attending college and working?


union of events

The union of two events, [latex]A[/latex] or [latex]B[/latex], is denoted by [latex]A \cup B[/latex].

 

The union of two events consists of the set of all elements in the collection of both events. The outcomes in the event [latex]A[/latex] or [latex]B[/latex] are the outcomes that are in event [latex]A[/latex], in event [latex]B[/latex], or in both event [latex]A[/latex] and event [latex]B[/latex].

 

In term of probability, [latex]P(A \text{ or }B)= P(A \cup B)=[/latex] the relative frequency of either event [latex]A[/latex] or [latex]B[/latex] (or both) with respect to the sample space.

Consider the set [latex]A = \{\text{red}, \text{blue}, \text{green}\}[/latex] and [latex]B=\{\text{red},\text{yellow},\text{orange}\}[/latex].

The event [latex]A \cup B = \{\text{red}, \text{blue}, \text{green}, \text{yellow}, \text{orange}\}[/latex].

Once again, we can utilize the Venn diagram to represent this as two events [latex]A[/latex] and [latex]B[/latex] with both circles (and their intersection) being entirely shaded as shown below.

This Venn diagram contains two overlapping circles within a rectangle labeled "S". The left circle is labeled "A" and the right circle is labeled "B". The region inside all of the circles is shaded to represent the Union.


mutually exclusive events

Mutually exclusive describes two or more events that cannot happen simultaneously.

 

If event [latex]A[/latex] and event [latex]B[/latex] are mutually exclusive, then [latex]A \cap B[/latex] is an empty set and [latex]P(A \cap B) = 0[/latex].

 

Therefore,

[latex]P(A \text{ or } B) = P(A \cup B) = P(A)+P(B)[/latex]

Using Venn diagrams, we can also visualize the concept of mutually exclusive or disjoint events as non-intersecting circles as shown below.

This diagram presents two non-overlapping circles within a rectangle labeled "S". The left circle is labeled "A", and the right circle is labeled "B".

When we find probability [latex]P(A \text{ or } B)[/latex] for mutually exclusive events, we would consider the shaded region of both circles without any intersection.