Uncertainty: Probability

Probabilities

Imagine an experiment in which we will select a single outcome (at random) from a space of possible outcomes. To each set of outcomes, A, which is a subset of the space of outcomes, we assign a number P(A) between 0 and 1 which represents the proportion of times the outcome would be in set A if we repeated the experiment many times. This number could either be an observed frequency or a subjective guess as to what the frequency would be if we could repeat the experiment many times. We place a rather strong restriction on the set of possible probabilities in that we assume that if A and B are disjoint sets of outcomes P(A or B) = P(A) + P(B). Thus probabilities are additive and we can uniquely define the probability distribution by assigning probabilities to each single outcome (or assigning probability density in the case that the outcome space is a dense set like the real numbers).

As an example, consider a valve which can be in one of two states: working or failed. Assume that it fails according to the rules of the Poisson process (that is it has constant failure rate) and that we have extensive test data which establishes the failure rate. Then, given the exposure time of the valve, we can calculate the probability that the valve will fail. Note that this calculation requires a exact knowledge about both the failure rate and the exposure times. Thus, we must be confident that both of those factors are well understood in order to use a first order probability model.

If our information is less precise, it is more difficult to build a probability model. There are two ways in which it could be imprecise: (1) if we don't know the failure rate (or other parameter) precisely, we can build a second order model by putting another probability distribution over the failure rate, and (2) our data could be imprecise. As an example of this latter, suppose that there are three different failure modes: stuck open, stuck closed, and ruptured, but our data only tells us the failure rate for failures of any mode. (This is actually quite common). Building a probability model for this situation requires assumptions about the relative likelihoods of the various failure states (for example, all failure states are equally likely). A generalization of probabilities, belief functions provide a model for this situation.

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