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| author | Gustav Sörnäs <gustav@sornas.net> | 2022-01-21 13:28:45 +0100 |
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| committer | Gustav Sörnäs <gustav@sornas.net> | 2022-01-21 13:28:45 +0100 |
| commit | c4950dfb23d1fe8cca7cb523d9ae4ea4b4dda2d5 (patch) | |
| tree | 991dd3862f8366a667de73aa03820f1701397e9a /sannolikhet/Event.md | |
| parent | e5aa9d5a17d52eb61243a9c45dd2e65f27084c9c (diff) | |
| download | notes-c4950dfb23d1fe8cca7cb523d9ae4ea4b4dda2d5.tar.gz | |
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diff --git a/sannolikhet/Event.md b/sannolikhet/Event.md new file mode 100644 index 0000000..1d630f0 --- /dev/null +++ b/sannolikhet/Event.md @@ -0,0 +1,54 @@ +An event is any subset of the [[Sample space]]. + +Example: Given +$$S = \{1,2,3,4,5,6\}$$ + +then +$$A = \{2,4\}$$ +is an event. + +Can be visualized with [[Venn diagram]]s. + +Events happen with a [[Probability]]. + +## Special events + +Some special events are the empty event ($\phi$) and the [[Sample space]] itself ($S$). + +# Operations + +Just like there are normal operations (+, \*, ..) on numbers there exist +operations on events. They work much like events on normal [[Set]]s. + +## Intersection + +$$A \cap B$$ + +## Union + +$$A \cup B$$ + +## Complement + +$$A'$$ + +# Disjoint + +Two events are disjoint if they don't "overlap" in any way. + +# Independent + +Two events are independent if $P(A \cap B) = P(A) \cdot P(B)$. + +Intuition: $A$ does not affect $B$ and vice versa. + +$(A, B)$ independent $\Leftrightarrow$ $(A', B)$ independent. + +For example, given a fair dice roll and + +$$A = \{2\}, \ B = \{2, 3\}, \ C = S$$ + +then $A$ and $B$ are dependent but $A$ and $C$ are independent (since the +outcome of $A$ doesn't affect the outcome of $C$ since $P(C) = 1$. + +In general, $A_1, ..., A_n$ are independent if... |
