Argument from Probability
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[edit] Definition
Argument from Probability is an informal logical fallacy where a participant infers that an unlikely event cannot occur, or alternatively that a likely event will always happen. The most general structure of this argument runs something like the following:
- The probability of P happening is <some small number>
- Therefore, not-P.
A simple variation on this is
- The probability of P not happening is <some small number>
- Therefore, P.
This is a fallacy because unlikely events can and do happen. This fallacy is often combined with mis-estimations of the probabilities involved or a confusion between prior and posterior probabilities, resulting in mathematically incorrect calculations.
[edit] Examples
Example 1:
- Antagonist: The chance of life arising by itself on this planet is a billion to one against; obviously life was created by another agency.
Example 2:
- Antagonist: The chances of the fingerprints matching were 100:1 against, so the defendant is 99% likely to be guilty.
[edit] Discussion
There are two closely related problems usually involved in fallacious arguments from probability. The first is simply the problem, often a difficult one, of getting the sums right and realizing exactly what the stated probability means. For example, in the second example above, the likelihood that a random person would match the fingerprints is not by itself proof of the defendant's guilt. If we have a pool of 200 suspects, we expect (by chance) two matches within that pool. If the guilty party is known to be in that pool, we can conclude that the non-matches are 0% likely to be guilty, but that both matches have approximately a 50/50 chance to be the guilty party. Similarly, a scientific study with a p-value of 1% does not mean that the hypothesis is 99% likely to be true, but that random data would have a 99% chance of getting a smaller effect than the experiment observed. To misinterpret a statement of probability in such a fashion is to commit the fallacy of equivocation.
One important and often-confused aspect of this is the difference in Bayesian probability theory between prior and posterior (formally, a priori and a posteriori) probabilities. The prior probability is the probability of an event occurring without taking evidence into account, while the posterior probability is the probability calculated when subsequent evidence is observed and analyzed. For example, it is highly improbable that the winning numbers on a lottery ticket would be 1,2,3,4,5 -- and if it were, people would suspect chicanery on the part of the lottery company. But it is exactly as unlikely that the winning numbers should be 4, 16, 24, 26, and 31. Before the lottery is held, all winning combinations are equally unlikely -- but after the lottery concludes, it is obvious that some unlikely pattern must have been the winning one.
For this reason, we cannot conclude that just because a proposed event that has already happened is unlikely, that the event itself did not occur and the proposed explanation is false. Similarly, in our first example, the fact that the origin of life might be wildly implausible does not in and of itself mean that it never happened. If life had not originated on our planet (through whatever means), no one would be here to discuss it on the JREF Wiki. Therefore improbability by itself cannot refute an explanation of an event that has already happened. As Sherlock Holmes put it, "Once you eliminate the impossible, whatever remains, no matter how improbable, must be the truth."
[edit] Exceptions to the Rule
Probability can be a legitimate tool for distinguishing between two different hypotheses. If we have two legitimate explanations for the same observation:
- My car doesn't start
- If space aliens stole the battery, it wouldn't start
- If I left the lights on last night, it wouldn't start
we can use a legitimate probability analysis to determine whether it's more likely that space aliens are interested in my car or that I am careless, and probably reject the first explanation.
