2.1.3.2.2 - Intersections

The term intersection is used to describe the overlap or two or more events. This is communicated using the character ∩. The phrase \(P(A \cap B)\) is read as "the probability of A and B."

In the form of a Venn diagram, we can picture this as the overlap between two [or more] events. 

Intersection of A and B

Example: Cards Section

What is the probability of randomly selecting a card from a standard 52-card deck that is a red card and a king?

There are 2 kings that are red cards: the king of hearts and the king of diamonds.

\(P(red \cap king)=\dfrac{2}{52}=.0385\)

Example: Penn State Enrollment Section

The two-way contingency table below displays the Penn State's undergraduate enrollments from Fall 2019 in terms of status (full-time and part-time) and primary campus (data from the Penn State Factbook).

  Full-Time Part-Time Total
University Park 39529 1110 40639
Commonwealth Campuses 24306 2794 27100
PA College of Technology 4110 871 4981
World Campus 2574 5786 8360
Total 70519 10561 81080

 

 

What proportion of Penn State students were full-time University Park students?

This is an example of a union because we are looking for the proportion of all students who are both full-time and University Park.

\(P(FullTime \cap UniversityPark)=\dfrac{39529}{81080}=0.488\)

 

What proportion of Penn State students were part-time World Campus students?

This is an example of a union because we are looking for the proportion of all students who are both part-time and World Campus.

\(P(PartTime \cap WorldCampus) = \dfrac{5786}{81080}=0.071\)