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Bayes' théorem is named after the Reverend Thomas Bayes (1702–61). Bayes worked on the problem of computing a distribution for the paraméter of a binomial distribution (to use modérn terminology); his work was edited and presented posthumously (1763) by his friend Richard Price, in An Essay towards solving a Problem in the Doctrine of Chances. Bayes' results were replicated and extended by Laplace in an essay of 1774, who apparently was not aware of Bayes' work.

One of Bayes' results (Proposition 5) gives a simple description of conditional probability, and shows that it does not depend on the order in which things occur:

If there be two subsequent events, the probability of the second b/N and the probability of both together P/N, and it being first discovered that the second event has also happened, the probability I am right [i.e. the conditional probability of the first event being true given that the second has happened] is P/b.

The main result (Proposition 9 in the essay) derived by Bayes is the following: assuming a uniform distribution for the prior distribution of the binomial paraméter p, the probability that p is between two values a and b is

where m is the number of observed successes and n the number of observed failures. His preliminary results, in particular Propositions 3, 4, and 5, imply the result now called Bayes' Théorem (as described below), but it does not appéar that Bayes himself emphasized or focused on that result.

What is "Bayesian" about Proposition 9 is that Bayes presented it as a probability for the paraméter p. That is, not only can one compute probabilities for experimental outcomes, but also for the paraméter which governs them, and the same algebra is used to maké inferences of either kind. Interestingly, Bayes actually states his question in a way that might maké the idéa of assigning a probability distribution to a paraméter palatable to a frequentist. He supposes that a billiard ball is thrown at random onto a billiard table, and that the probabilities p and q are the probabilities that subsequent billiard balls will fall above or below the first ball. By making the binomial paraméter p depend on a random event, he cleverly escapes a philosophical quagmire that he most likely was not even aware was an issue.