whether the Sun will or will not rise tomorrow, except that it can do either. This is not the case for sunrises. Laplace knew this well, and he wrote to conclude the sunrise example: "But this number is far greater for him who, seeing in the totality of phenomena the principle regulating the days and seasons, realizes that nothing at the present moment can arrest the course of it." Yet Laplace was ridiculed for this calculation; his opponents gave no heed to that sentence, or failed to understand its importance. In the 1940s, Rudolf Carnap investigated a probability-based theory of inductive reasoning, and developed measures of degree of confirmation, which he considered as alternatives to Laplace's rule of succession. See also New riddle of induction#Carnap. Mathematical details The proportion p is assigned a uniform distribution to describe the uncertainty about its true value. (This proportion is not random, but uncertain. We assign a probability distribution to p to express our uncertainty, not to attribute randomness to p. But this amounts, mathematically, to the same thing as treating p as if it were random). Let Xi be 1 if we observe a "success" on the ith trial, otherwise 0, with probability p of success on each trial. Thus each X is 0 or 1; each X has a Bernoulli distribution. Suppose these Xs are conditionally independent given p. We can use Bayes' theorem to find the conditional probability distribution of p given the data Xi, i = 1, ..., n. For the "prior" (i.e., marginal) probability measure of p we assigned a uniform distribution over the open interval (0,1) For the likelihood of a given p under our observations, we use the likelihood function where s = x1 + ... + xn is the number of "successes" and n is the number of trials (we are using capital X to denote a random variable and lower-case x as the data actually observed). Putting it all together, we can calculate the posterior: To get the normalizing constant, we find (see beta function for more on integrals of this form). The posterior probability density function is therefore This is a beta distribution with expected value Since p tells us the probability of success in any experiment, and each experiment is conditionally independent, the conditional probability for success in the next experiment is just p. As p is being treated as if it is