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Michael O. Rabin analyzed a randomized algorithm for primality testing in his 1980 paper. The Miller-Rabin primality test checks whether a candidate integer satisfies conditions obeyed by primes under modular exponentiation. A failed check certifies compositeness; passing a check gives a probable prime, not a proof of primality.^[ref]

For each fixed odd composite candidate, at most one quarter of the admissible bases are strong liars. Independent uniformly chosen bases therefore reduce the probability of accepting that composite to at most $4^{-k}$ after $k$ rounds. This probability is over the test's choices for a fixed input; it is not a posterior probability that an arbitrary accepted integer is prime.

A probable prime is an integer that passes a specified necessary test for primality without thereby receiving a proof of primality. The qualification must identify the test and bases. For example, an odd $n$ passing a Fermat test to base $a$ satisfies $a^{n-1}\equiv 1(\mathrm{mod}n)$, but a composite can satisfy that congruence.

The Miller-Rabin primality test imposes a stronger condition on a chain of modular squares. Yet $2047=23\cdot 89$ passes even that test to base 2: writing $2046=2\cdot 1023$, one has $2^{1023}=(2^{11}{)}^{93}\equiv 1(\mathrm{mod}2047)$. A pass therefore means “not disproved by this test”, whereas a failing witness proves compositeness. With independent uniformly sampled bases, the standard worst-case error bound decreases exponentially with the number of rounds; it is a conditional test-error bound, not automatically the posterior probability that the particular input is prime.^[ref]