Type: Formal Fallacy
A probabilistic argument is one which concludes that something has some probability based upon information about probabilities given in its premisses. Such an argument is invalid when the inference from the premisses to the conclusion violates the laws of probability. Probabilistic fallacies are formal ones because they involve reasoning which violates the formal rules of probability theory. Thus, understanding probabilistic fallacies requires a knowledge of probability theory.
A Short Introduction to Probability Theory:
In the following laws of probability, the probability of a statement, s, is represented as: P(s).
The above laws are logically sufficient to prove every fact within probability theory, including a theorem that is important for explaining probabilistic fallacies:
Bayes' Theorem: P(s | t) =
P(t | s)P(s) / [P(t | s)P(s) + P(t | not-s)P(not-s)].
Proof: From axiom 4, we know that P(s | t) = P(s & t)/P(t). Since "s & t" is logically equivalent to "t & s", P(s & t) = P(t | s)P(s), again by axiom 4, which is the numerator of the fraction in Bayes' Theorem. To get the denominator of the fraction, "t" is logically equivalent to "(t & s) or (t & not-s)", so P(t) =
Mistakes in reasoning about probabilities are typically not treated as formal fallacies by logicians. This is presumably because logicians usually do not make a study of probability theory, and the mathematicians who do don't generally study logical fallacies. However, in recent decades, psychologists have discovered through observation and experiment that people are prone to make certain types of error when reasoning about probabilities. As a consequence, there is now much more empirical evidence for the existence of certain fallacies about probabilities than there is for most traditional fallacies. Again, logicians are often unaware of the existence of this evidence, and they usually do not discuss it in works on logical fallacies. It is about time that logicians broadened their intellectual horizons and began to take note of discoveries in the psychology of reasoning.
Resource: Amir D. Aczel, Chance: A Guide to Gambling, Love, the Stock Market, & Just About Everything Else (2004). About as untechnical an introduction to probability theory as you will find.
Acknowledgment: Thanks to Emil William Kirkegaard for pointing out a problem, which has subsequently been fixed, with the wording of the informal description of the first axiom of probability.