5.2: Known Outcomes

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Page ID: 50183

Paul Penfield, Jr.
Massachusetts Institute of Technology via MIT OpenCourseWare

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Once you know an outcome, it is straightforward to denote it. You merely specify which symbol was selected. If the other events are defined in terms of the symbols, you then know which of those events has occurred. However, until the outcome is known you cannot express your state of knowledge in this way. And keep in mind, of course, that your knowledge may be different from another person’s knowledge, i.e., knowledge is subjective, or as some might say, “observer-dependent.”

Here is a more complicated way of denoting a known outcome, that is useful because it can generalize to the situation where the outcome is not yet known. Let i be an index running over a partition. Because the number of symbols is finite, we can consider this index running from 0 through \(n−1\), where \(n\) is the number of events in the partition. Then for any particular event \(A_i\) in the partition, define \(p(A_i)\) to be either 1 (if the corresponding outcome is selected) or 0 (if not selected). Within any partition, there would be exactly one \(i\) for which \(p(A_i)\) = 1 and all the other \(p(A_i)\) would be 0. This same notation can apply to events that are not in a partition—if the event \(A\) happens as a result of the selection, then \(p(A)\) = 1 and otherwise \(p(A)\) = 0.

It follows from this definition that \(p(\text{universal event})\) = 1 and \(p(\text{null event})\) = 0.