8.2.2: Probabilities

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Page ID: 51672

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

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This example has been defined so that the choice of one of the three meals constitutes an outcome. If we do not know this outcome we may still have some knowledge, and we use probabilities to express this knowledge. The question is how to assign probabilities that are consistent with whatever information we may have.

In the case of Berger’s Burgers, there are three probabilities which for simplicity we denote \(p(B)\), \(p(C)\), and \(p(F)\) for the three meals. A probability distribution \(p(A_i)\) has the property that each of the probabilities is between or equal to 0 and 1, and, since the input events are mutually exclusive and exhaustive, the sum of all the probabilities is 1:

\(\begin{align*} 1 &= \displaystyle \sum_{i} p(A_i) \\ &= p(B) + p(C) + p(F) \end{align*} \tag{8.13}\)

If any of the probabilities is equal to 1 then all the other probabilities are 0, and we know exactly which state the system is in; in other words, we have no uncertainty and there is no need to resort to probabilities.