Appeal to probability

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The appeal to probability is a logical fallacy. It assumes that because something could happen, it is inevitable that it will happen. This is flawed logic, regardless of the likelihood of the event in question. The fallacy is often used to exploit paranoia.

Form

This has the argument form:

Possibly P.

Therefore, P.

Equivalently, using modal logic and logical connective notation:

→ P

Examples

Some examples are:

• "There are many hackers that spread worms through the internet. Therefore, if you use the internet without a firewall, it is inevitable that you will be hacked sooner or later."
  • While using a firewall is a prudent and sensible measure, it is not inevitable that a hacker will attack an unprotected computer. The argument does have some backing logic, especially in the case of Warhol worms designed to reach almost every vulnerable computer in minutes, but overstates the worst case scenario.
• "It doesn't matter if I get myself into debt. If I play the lottery, I might win the jackpot, and then I can pay off all my debts."
  • A reversal of the previous argument, as it is dependent on the best case scenario coming true. This is a dangerous argument, hinged on the relatively small (usually in the thousands or millions to one) odds of winning the jackpot, or at least enough to make a large enough difference to someone's debts, and then using these small odds to justify excessive amounts of debt. In fact, a lottery ticket often has an expected value close to half the ticket price.
• "When soccer becomes popular in a town, hooliganism will become a major problem. Thus, if we allow a soccer team in our town, we will be overrun by hooligans."
  • This is another argument that assumes a worst-case scenario, without any other backing logic. In addition, it also falsely implies that correlation means causation, as it assumes that soccer is the direct cause of hooliganism, without taking into account other socio-economic factors.

Explanation

^{[clarification needed]}The logical idea behind this fallacy is usually that, if the probability of P occurring is approaching 1, it is best to assume that P will occur, since it will almost surely happen. The fallacy incorrectly applies a common tenet of probability: given a sufficiently large sample space, an event X of nonzero probability P(X) will occur at least once, regardless of the magnitude of P(X). This is derived from the definition of probability. The operative term is "given a sufficiently large sample space". Virtually all events are considered for probability within a finite number of samples, and the chance that X will occur in a given finite space S is directly proportional to S. Given a finite number of events S, each of which is X or not X, a sample space Y = ₂Pr_S exists where one possibility is that all events in S are not X. Therefore, P(X in Y) = (Y-1)/Y. Because Y-1/Y < 1 for all finite Y, P(X in Y) < 1 regardless of P(X) or Y. There is thus always a chance that X will not occur, and therefore, no proof that X will occur given its probability.

See also

• Murphy's law