# Lecture 1. A) Sample Space

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## Sample Space

Traditionally, in probability theory, tossing a coin can yield only Heads or Tails. The set of all possible cases, Heads or Tails in this case, is called the sample space. (It’s the space from which one can sample from, for example, when one runs an experiment, or essentially when one performs a measurement, like observing the outcome of tossing a coin in the air.)

Let's introduce some more notation:
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• $\emptyset$ is the empty set.

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• $\bigcup_{i=1}^{\infty}B_{i}$ is the union of sets $B_{i}$, formally, $\bigcup_{i=1}^{\infty}B_{i}=\left\{ s\in S:s\in B_{i}\,for\,some\,i\right\}$

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• $B\subseteq S$ means $B$ is a subset of the sample space. For example, $B=\left\{ Heads\right\}$ is a subset of $S=\left\{ Heads,Tails\right\}$.

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• From the line above, notice that $Heads$ is an element of set $B$ (and $S$). Notice the absence of curly braces.

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• $B^{C}=S\backslash B$ is the complement of set $B$. When $B=\left\{ Heads\right\}$, $B^{C}=\left\{ Tails\right\}$. Formally, $B^{C}=\left\{ s\in S:s\notin B\right\}$.