6 ideas
Earlier probability theory ran forward, from a known chance to the likely outcomes of trials. Bayes reverses the direction. Given only the observed count of successes and failures, he computes the probability that the unknown underlying chance lies between any two chosen bounds. This makes the cause the uncertain quantity and treats the data as fixed.
A first ball is rolled at random onto a level table, and its unseen position fixes an unknown chance. A second ball is then rolled repeatedly, and each roll counts as a success if it stops on one side of the first ball. Because the first ball's position is equally likely anywhere, the setup gives a physical justification for a uniform prior. It lets the posterior probability be computed exactly as a ratio of areas.
The probability of a hypothesis after seeing evidence is proportional to two things multiplied together. The first is how probable the hypothesis was beforehand. The second is how probable the evidence would be if the hypothesis were true. The result is then normalized across all competing hypotheses, so every new observation reweights the old belief rather than replacing it.
When we know nothing about an event before any trials, we have no reason to think one number of successes more likely than another. Bayes argues that this state of ignorance warrants treating every possible value of the unknown chance as equally likely beforehand.
Bayes defines the probability of an event as the ratio between the value of an expectation that depends on the event happening and the value of the thing expected if it happens. This roots probability in what a reasonable person should expect or pay. It does not depend on long-run frequencies, so probability can apply to single, unrepeatable questions about unknown causes.
Price's supplement applies the method to someone who watches the sun rise repeatedly. That observer's confidence in a steady underlying regularity grows with each confirmation but never reaches certainty. Seen this way, induction from experience becomes a quantifiable degree of belief that tightens as evidence accumulates, which answers the problem of induction with calculation rather than proof.