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For example, suppose that I perform an experiment

If the experiment is repeated infinitely many times, then the frequency of occurrences of A should be the probability of A; the probability of A is the limiting frequency of occurrences of A. This is the

Some authors use probability to reflect their subjective degree of belief in the truth of statements. This is a different use of the formalism of probability theory.

Probabilities are numbers assigned to events, i.e. subsets of the sample space (for our purposes). (When dealing with certain infinite sample spaces, not just any subset of the sample space can be allowed to be an event. This fact will not be needed for this class.) The axioms of probability were inspired by the ways that relative frequencies behave; there will never be a negative probability, nor will probabilities ever be greater than one.

The first axiom is that the probability is a number between 0 and 1 which is assigned to certain subsets of the sample space, called events. It is standard to write P(A) to denote the probability of the event A.

The next is that the probability the sample space U itself, P(U), always equals 1. The sample space U may be called the certain event.

Remember that probability is meant to be the limiting frequency of occurrences of an event as the experiment is repeated more and more times. The sample space is the collection of all the outcomes of the event. Suppose I consider the event that something in the sample space occurs. For instance, suppose I roll a single die over and over again, and I ask how often I see either a 1, 2, 3, 4, 5, or a 6. Since the sample space consists of these and only these possibilities, every time I roll the die, I see one of them. The frequency is 1, always. For this reason, the probability of the sample space itself (taken as an event) is defined to be 1. This is written

The final axiom involves disjoint unions. Suppose you have a disjoint sequence of events like A

On to probabilities of complementary events.

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