2.1.3.2.3  Unions
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
What is the probability of randomly selecting a card from a standard 52card 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
What is the probability of randomly selecting a card from a standard 52card 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
The twoway 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:
\(P(F \cup U) = \dfrac{6027}{12242}+\dfrac{7242}{12242}\dfrac{3814}{12242}= \dfrac{9455}{12242}=0.772\)