6 ideas
An intellect that knew, at one instant, every force acting in nature and the position of every body, and could analyze these data, would see the future and past as clearly as the present. Probability therefore exists not in the world but in our partial ignorance of causes, and it measures what we know against what we lack.
The probability of an event is the number of favorable cases divided by all possible cases, provided we have equal grounds for uncertainty about each case. When cases are not equally possible, their relative possibilities must be determined first, and this judgment about equipossibility carries the real weight of the calculation.
The probability that a witness is right depends both on the witness's reliability and on how improbable the reported event is in itself. As an event becomes more extraordinary, the chance the witness erred or lied becomes larger than the chance the event happened. So miraculous claims require evidence strong enough to outweigh their prior improbability, and chains of transmitted testimony lose credibility with each added link.
When an observed event could arise from several causes, the probability of each cause is proportional to its prior probability times the probability that it would produce the event, so observation revises beliefs about hidden causes. Laplace applies this to an event of fixed but unknown probability, first treating all possible values of that probability as equally likely. On those assumptions, after the event has occurred n times without failure, the probability it occurs next is (n+1)/(n+2).
People wrongly believe a number that has not come up for a long time in a lottery is more likely to appear soon, as if past draws influenced independent future ones. Laplace groups this with other psychological illusions in estimating probability, where hope, fear, and memory of striking coincidences distort judgment away from what calculation shows.
The value of a gain to a person depends on how it compares to their existing fortune, so moral expectation differs from the plain mathematical expectation of money. Since each added unit of wealth matters less, even fair games are disadvantageous to players, while spreading risk through insurance or diversification raises moral expectation for the same mathematical value.