# What is the meaning of $P(A \cup B)$ in probability and statistics?

In probability, letters $A$ and $B$ are used to denote various events. Then we write $P(A)$ for the probability of event $A$ happening. Same for $P(B)$.

But I often see the notation $P(A\cup B)$ as well. What does it mean, and how is such a thing calculated?

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Do you remember studying sets? This is the probability of the event $A\cup B$, the union of events $A$ and $B$. It represents the probability of $A$ or $B$. – Neal Sep 23 '12 at 13:39

$P(A\cup B)$ is simply the probability that at least one of $A$ and $B$ occurs. $\cup$ is the symbol for set union, and events in probability theory are described by sets.

For example, take throwing a die. Take $A$ to be the event "an even number was thrown", represented by the set $\{2,4,6\}$ and $B$ to be the event "a prime number was thrown", represented by the set $\{2,3,5\}$. Then $A\cup B$ is the event "an even number or a prime number was thrown", that is, the union $A\cup B=\{2,3,4,5,6\}$.

Then $P(A\cup B)$ is the probability that you've thrown an even or a prime number, that is the probability that your result was one of the numbers from $2$ to $6$.

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If $A,B$ are two events ,then,

$P(A\cup B)$ represents the probability of happening atleast one of the event($A$ or $B$).

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Suppose that there are $n$ events possible. Let $A$ be a set of $n - k$ arbitrary events and $B$ the set of other $k$ events. Then $\rm A \cup B$ denotes all the events possible.

$P({\rm A \cup B})$ denotes the probability of an event happening from either of set $\rm A$, or $\rm{B}$ (which would be $1$ in this case since all events possible are contained within $\rm A \cup B$).

Example. Suppose we are throwing a fair dice. Let $\rm O$ denote the set of events where the number we get is odd. Let $\rm E$ denote the set of events where the number we get is even. It is obvious that,\begin{aligned} \rm O &= \{1,3,5\} \\ \rm E &= \{2,4,6\} \\\rm O \cup E & = \{1,2,3,4,5,6\} \end{aligned}

Property. If $\rm A \cap B = \emptyset$, then $\rm P(A \cup B) = 1$ (if $A$ and $B$ are the only sets).

Pop Quiz.

• What do you infer from $\rm P(A \cap B)$?
• Can you find other properties for different cases?
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is there something wrong with my eyes or does B appear larger than A for some reason? – Elements in Space Dec 26 '12 at 13:51