Sample Spaces and Events

Imagine performing an experiment with a set of possible outcomes. Initially, the result is unknown. After performing the experiment, an outcome is observed.

$\textbf{Definition}$ (Sample Space). The $\textit{sample space S}$ of an experiment is the set of all possible outcomes of the experiment. [1]

$\textbf{Definition}$ (Event). An $\textit{event A}$ is a subset of the sample space $\textit{S}$, and we say that $\textit{A}$ occurred if the actual outcome is in $\textit{A}.$ [1]

$\textbf{Example}$ (Tossing a coin two times). Suppose you toss a coin two times. The sample space is $S = \left\{HH, HT, TH, TT\right\}$, where $\textit{H}$ denotes heads and $\textit{T}$ - tails. Consider the following events:

  • The event $A$ that the result of the first toss was heads.
  • The event $B$ that the result of the first toss was tails.
  • The event $C$ that the outcomes of both tosses were heads.
  • The event $D$ that the outcome of the second toss was the same as the outcome of the first toss.

All four events are subsets of the sample space. For each of them we have:

  • $A = \left\{HH, HT\right\}.$
  • $B = \left\{TH, TT\right\}.$
  • $C = \left\{HH\right\}.$
  • $D = \left\{HH, TT\right\}.$

Probability theory is build upon set theory as the latter provides a very useful framework for working with events. For example, let $S$ be the sample space of an experiment and let $A, B\subset S$ be events. Then the event $A\cap B$ is the event that both $A$ and $B$ occurred, $A^{c}$ occurs if and only if $A$ didn't occur and $A\cup B$ is the event that either $A$ or $B$ occurred. We can use the important set relation results from set theory to analyze events, such as, for instance, De Morgan's Laws: \begin{equation} \left(A\cup B\right)^{c} = A^{c}\cap B^{c} \end{equation} \begin{equation} \left(A\cap B\right)^{c} = A^{c}\cup B^{c} \end{equation} Here the first equation tells us that the event that $A$ or $B$ didn't happen is the same as the event that both $A$ didn't happen and $B$ didn't happen. The second equation states that the event that both $A$ and $B$ didn't happen is the equivalent to $A$ didn't happen or $B$ didn't happen. Depending on the situation, one way of writing out an event might be more convenient than the other.

References

[1]    J. K. Blitzstein and J. Hwang, Introduction to Probability /, Second edition. (2019).