3.1: Events

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Page ID: 47232

Franz S. Hover & Michael S. Triantafyllou
Massachusetts Institute of Technology via MIT OpenCourseWare

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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.

Visual representation of simple events (left) vs composite events (right) — Figure \(\PageIndex{1}\): A visual representation of simple events (left), where \(A_1\) through \(A_5\) do not overlap each other, compared to composite events (right), where events \(A_1\) through \(A_5\) overlap so that a sample may invoke multiple events.

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.

Visual representation of an event M that contains multiple simple events

\(\PageIndex{2}\): A visual representation of an event space completely occupied by simple events \(A_i\), and an event M that incorporates elements of multiple possible \(A_i\).