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# Probability

Probability is a very important concept in the study of inductive logic. In probability theory, we deal with sentences that are, in some sense, "probable." That is, we deal with sentences whose truth values we are uncertain of. Will it rain today? Should I hit or stand on a 16 in Blackjack? Do I think Frank is the murderer? These are all questions that importantly involve probability.

The word "probability" has many different meanings. We say things like, "the probability of rain is 60%" or "the probability of me winning this hand of cards is very, very low." In an attempt to clarify and explain the different ways that "probability" is used, we provide different interpretations of probability.

We express the probability of a sentence A as "Pr(A)"

In order to treat probability in a rigorous and formal way, we treat it as a mathematical function that assigns numbers to sentences. This function intends to capture the relevant facts of how probability works; to this end we give a set of rules that defines how this function works. By using these mathematical tools, we can give definitive answers to problems in probability.

If these tools are too formal for you to understand, probability spaces are a helpful way to picture (literally) what the concepts of probability theory mean.

Probability is also important to you, as a person who believes things and acts on the basis of those beliefs. One of the important interpretations of probability is as a degree of belief, a measure of how strongly you believe something (or how strongly you should believe something). But why should your beliefs follow these mathematical rules? A strong answer to this question is given by the Dutch Book arguments, which show what can go wrong if you don't follow these rules.

Probability's connection to belief also means it has an important connection to action. This connection is given by expected utility theory.

Another important application of probability is in the method of hypothesis and deduction, which is the principle method of science. Science proceeds by formulating hypotheses, gathering evidence to test those hypotheses, then revising them on the basis of successful or failed experiments. The formal tools of probability theory show not only how this method works, but why this is the correct method in learning about the world.