# 2.1.3.2.3 - Unions

Union

A union contains the area in A or B and is symbolized by $$\cup$$. Note that this also includes the overlap of A and B (i.e., the intersection).

$$P(A \cup B)$$ is read as "the probability of A or B."

Union
$$P(A\cup B) = P(A)+P(B)-P(A\cap B)$$

## Example: Hearts or Spades Section

What is the probability of randomly selecting a card from a standard 52-card deck that is a heart or spade?

There are 13 cards that are hearts, 13 cards that are spades, and no cards that are both a heart and a spade.

$$P(heart \cup spade)=\dfrac{13}{52}+\dfrac{13}{52}-\dfrac{0}{52}= \dfrac {26}{52}=0.5$$

## Example: Hearts or Aces Section

What is the probability of randomly selecting a card from a standard 52-card deck that is a heart or an ace?

There are 13 cards that are hearts and 4 cards that are aces. There is one ace of hearts, so one of those 4 aces has already been counted.

$$P(heart \cup ace)=\dfrac{13}{52}+\dfrac{4}{52}-\dfrac{1}{52}=\dfrac{16}{52}=0.308$$

## Example: Female or Undergraduate Section

The two-way table below displays the World Campus enrollment from Fall 2015 in terms of level (undergraduate and graduate) and biological sex. What proportion of World Campus students were female or undergraduate students?

Female Male Total
Undergraduate 3814 3428 7242
Graduate 2213 2787 5000
Total 6027 6215 12242

When we have a contingency table we can take the appropriate values from the table as opposed to using the formula given above. There are 3814 female undergraduate students, 3428 male undergraduate students, 2213 female graduate students, and a total of 12242 students.

$$P(F \cup U)=\dfrac{3814+3428+2213}{12242}=\dfrac{9455}{12242}=0.772$$

Note that the final answer would be the same if we had used the formula:

$$P(F \cup U) = \dfrac{6027}{12242}+\dfrac{7242}{12242}-\dfrac{3814}{12242}= \dfrac{9455}{12242}=0.772$$