5.3: Unknown Outcomes

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Page ID: 50184

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

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If the symbol has not yet been selected, or you do not yet know the outcome, then each \(p(A)\) can be given a number between 0 and 1, higher numbers representing a greater belief that this event will happen, and lower numbers representing a belief that this event will probably not happen. If you are certain that some event \(A\) is impossible then \(p(A)\) = 0. If and when the outcome is learned, each \(p(A)\) can be adjusted to 0 or 1. Again note that \(p(A)\) depends on your state of knowledge and is therefore subjective.

The ways these numbers should be assigned to best express our knowledge will be developed in later chapters. However, we do require that they obey the fundamental axioms of probability theory, and we will call them probabilities (the set of probabilities that apply to a partition will be called a probability distribution). By definition, for any event \(A\)

\(0 \leq p(A) \leq 1 \tag{5.1}\)

In our example, we can then characterize our understanding of the gender of a freshman not yet selected (or not yet known) in terms of the probability \(p(W)\) that the person selected is a woman. Similarly, \(p(CA)\) might denote the probability that the person selected is from California.

To be consistent with probability theory, if some event \(A\) happens only upon the occurrence of any of certain other events \(A_i\) that are mutually exclusive (for example because they are from a partition) then \(p(A)\) is the sum of the various \(p(A_i)\) of those events:

\(p(A) = \displaystyle \sum_{i} p(A_i) \tag{5.2}\)

where \(i\) is an index over the events in question. This implies that for any partition, since \(p(\text{universal event})\) = 1,

\(1 = \displaystyle \sum_{i} p(A_i) \tag{5.3}\)

where the sum here is over all events in the partition.