Encyclopedia>Statistics and Probability>Probability and Combinatorics>Basic Probability>What Are Disjoint Events in Probability?

Theory

Disjoint events are events that never occur together. Two events $A$ and $B$ are disjoint if the intersection of the sets is empty. Mathematically, this is written as

$$\begin{array}{llll}\hfill P\phantom{\rule{-0.17em}{0ex}}\left(A\cap B\right)& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \text{because}A\cap B& =\text{noelements.}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

$$P\phantom{\rule{-0.17em}{0ex}}\left(A\cap B\right)=0\text{because}A\cap B=\text{noelements.}$$ |

Example 1

You roll a die. Let $A$ be the compound event defined as “more than four dots”, $A=\phantom{\rule{-0.17em}{0ex}}\left\{5,6\right\}$, and $B$ be “fewer than three dots”, $B=\phantom{\rule{-0.17em}{0ex}}\left\{1,2\right\}$. That means the intersection between $A$ and $B$ is the empty set, because they have no outcomes in common. They can never both occur at once.

You denote the empty set with the symbol $\varnothing $.

**Note!** Complementary outcomes are always disjoint, because an event can not both happen and not happen at the same time.

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