Probability Theory Website
What did Thomas Bayes really say in An Essay towards solving a Problem in the Doctrine of Chances (1763)?
We all use “Bayes’s Rule” to calculate the posterior probability of A given B from the probability of B given A, the prior probability of A, and the overall probability of B:
$$P(A|B) = \frac{P(B|A)P(A)}{P(B)} $$But Bayes never wrote it.
Instead, he said something more specific about the probability of 
Here is how he said it in words:
… in the case of an event concerning the probability of which we absolutely know nothing antecedently, …I have no reason to think that, in a certain number of trials, it should rather happen any one possible number of times than another.
And here is my “translation”:
In independent trials of an unknown binary event, all possible success counts are equally likely.
The mathematical rule that he highlighted was, in modern notation,
$$ \int_{0}^{1} \binom{n}{k} \theta^{k}(1-\theta)^{n-k}d\theta = \frac{1}{n+1} \quad k = 0, 1, 2, …n.$$The main purpose of this website is to present my article, What Did Bayes Really Say? and provide links to my two other websites: EBD-2.net and sample-size.net . EBD-2.net is the website for the textbook Evidence-Based Diagnosis, 2nd edition, by Thomas B. Newman and Michael A. Kohn, illustrated by Martina A. Steurer. Sample-size.net is a website of calculators for clinical investigators. It was was developed by Michael Kohn and Josh Senyak (Quicksilver Consulting).
— Michael Kohn, updated 7 September 2022