- 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
Venn diagrams can also be useful when trying to visualize conditional probability.
Conditional probability is a way to calculate the likelihood of an event happening, considering that another event has already occurred. It’s like adjusting the probability based on additional information you have.
The conditional probability of [latex]A[/latex] given [latex]B[/latex], denoted as [latex]P(A\text{ given }B) = P(A|B)[/latex], represents the represents the probability of event [latex]A[/latex] occurring given that event [latex]B[/latex] has occurred.
The formula is given by:
[latex]P(A|B) = \dfrac{\text{Number of times both A and B occur}}{\text{Number of times B occurs}}[/latex]
OR
[latex]P(A|B) = \dfrac{P(A \cap B)}{P(B)}[/latex]
Where:
- [latex]P(A|B)[/latex] is the condition probability of event [latex]A[/latex] occurring given event [latex]B[/latex] has occurred.
- [latex]P(A\text{ and }B)=P(A \cap B)[/latex] is the probability of both events occurring
- [latex]P(B)[/latex] is the probability of event [latex]B[/latex] occurring.
Suppose in a study was conducted on pet ownership. Out of the [latex]50[/latex] participants, the study discovered that there were [latex]40\%[/latex] cat owners and [latex]60\%[/latex] were dog owners. Furthermore, [latex]10\%[/latex] of the participants were both cat and dog owners.
Let [latex]C[/latex] be the event someone is a cat owner and [latex]D[/latex] be the event someone is a dog owner.
(a) Construct the Venn diagram for the given scenario with the counts for the different events.
(b) How many people are cat owners?
(c) How many people are in this study are cat or dog owners?
(d) How many cat owners own a dog?
(e) Given that a person is cat owner, what is the probability they are also a dog owner?