Ah, right. The example given by Fallis is Rabin’s method for probabilistically proving that a number is prime. Some theorem is established showing that for any n, if it is not prime, then at least 3/4 of the numbers less than n have some property that easily verifies that n is composite. By checking 10 such numbers pseudorandomly, we can show that n is prime with error probability only 1 in a million. There’s a chance of error, but it can easily be made very low – much lower than the chance of error in a long deductive proof of primality by trial division. (Technically speaking, this just means we can reduce P(positive|false), rather than P(false|positive) – that is, we reduce the likelihood, not the posterior. To calculate the posterior, we need to know the prior probability, which we can’t say much about, except that it might be on the order of 1/log n.)
I never learned too many Erdös-style proofs using the probabilistic method in graph theory, but I recall that all the ones I learned could have been phrased entirely as counting arguments rather than involving probabilities. I was told that some more complicated ones couldn’t be, but I haven’t seen any like that myself. Of course, all these arguments are perfectly deductively valid, despite the misleading name.

Note that Fallis doesn’t include verification of the Goldbach Conjecture for many different even numbers. This is because we have no idea what the distribution of counterexamples would be like if there were counterexamples. Thus, in this case we can’t even get the likelihood, let alone the posterior, so we can’t give any sort of relevant probabilistic argument.

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