3.1: Events
- Page ID
- 47232
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Define an event space \(S\) that has in it a number of events \(A_i\). If the set of possible events \(A_i\) covers the space completely, then we will always get one of the events when we take a sample. On the other hand, if some of the space \(S\) is not covered with an \(A_i\) then it is possible that a sample is not classified as any of the events \(A_i\). Events \(A_i\) may be overlapping in the event space, in which case they are composite events; a sample may invoke multiple events. But the \(A_i\) may not overlap, in which case they are simple events, and a sample brings only one event \(A_i\), or none if the space \(S\) is not covered.
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Intuitively, the probability of an event is the fraction of the number of positive outcomes to the total number of outcomes. Assign to each event a probability, so that we have
\begin{align} p_i &= p(A_i) \geq 0 \\[4pt] p(S) &= 1 \end{align}
That is, each defined event \(A_i\) has a probability of occurring that is greater than zero, and the probability of getting a sample from the entire event space is one. Hence, the probability has the interpretation of the area of the event \(A_i\). It follows that the probability of \(A_i\) is exactly one minus the probability of \(A_i\) not occurring:
\[ p(A_i) = 1 - p(\bar{A}_i). \]
Furthermore, we say that if \(A_i\) and \(A_j\) are non-overlapping, then the probability of either \(A_i\) or \(A_j\) occurring is the same as the sum of the separate probabilities:
\[ p(A_i \cup A_j) = p(A_i) + p(A_j). \]
Similarly, if the \(A_i\) and \(A_j\) do overlap, then the probability of either or both occurring is the sum of the separate probabilities minus the sum of both occurring:
\[ p(A_i \cup A_j) = p(A_i) + p(A_j) - p(A_i \cap A_j). \]
As a tangible example, consider a six-sided die. Here there are six events \(A_1, A_2, A_3, A_4, A_5, A_6,\) corresponding with the six possible values that occur in a sample, and \(p(A_i) = 1/6\) for all \(i\). The event that the sample is an even number is \(M = A_2 \cup A_4 \cup A_6\), and this is a composite event.
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\(\PageIndex{2}\): A visual representation of an event space completely occupied by simple events \(A_i\), and an event M that incorporates elements of multiple possible \(A_i\).