## Laws Of Probability

The three laws of probability can be summarized as follows (see Evett and Weir 1998). First, as stated earlier, probabilities can take place in the range zero to one. Events that are certain have a probability of one while those that are not possible have a probability of zero. Thus, if a proposition or possibility is false, it has a probability of zero.

Second, events can be mutually exclusive meaning that if any one of a particular set of events has occurred then none of the others has occurred. If two events are mutually exclusive and we wish to know the probability that one or other of them is true then we can simply add their probabilities. This concept can be written out in the form:

or verbally, the probability of events G or H occurring given evidence E is equal to the probability of event G occurring given evidence E plus the probability of event H occurring given evidence E. In this example, all possibilities are captured by events G or H. Thus, if event G occurred then event H did not and visa versa. Another way to write this concept is that P(G|E) + P(H|E) = 1 and therefore upon rearranging the equation P(H | E) = 1 — P(G | E). This then means that the probability that H is false is equal to one minus the probability that H is true.

The third law of probability centers on the fact that when two events are independent of one another their probabilities can be multiplied with one another.

or verbally, the probability of events G and H occurring given evidence E is equal to the probability of event G given evidence E multiplied by the probability of event H given event G and evidence E.

If the conditioning information (evidence E) is clearly specified and consistent for all possible events, then we can drop the '|E' or 'given evidence E' portion of the equation to arrive at:

And if G and H are statistically independent or unassociated events then: P(G and H) = P(G) xP(H)

To summarize, probabilities fall in the range of 0 to 1. When considering the possibilities of two events occurring, if either one of two mutually exclusive events can occur, their individual probabilities are added (sum rule). Alternatively, if we wish to consider the probability of two independent events occurring simultaneously, then the individual probabilities can be multiplied (product rule). 