In probability theory, the rule of succession is a formula introduced in the 18th century by Pierre-Simon Laplace in the course of treating the sunrise problem. Probability theory is the mathematical study of probability. ... Pierre-Simon Laplace. ... The sunrise problem can be expressed as follows : What is the probability that the sun will rise tomorrow ? The sunrise problem illustrates the difficulty of using probability theory when evaluating the plausibility of statements or beliefs. ...

Statement of the rule of succession

Suppose p is uniformly distributed on the interval [0, 1]. Suppose X₁, ..., X_n+1 are conditionally independent random variables given the value of p, and conditional on p are Bernoulli-distributed with expected value p, i.e., each has probability p of being equal to 1 and probability 1 − p of being equal to 0. Then In mathematics, the uniform distributions are simple probability distributions. ... In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Two random variables X and Y are conditionally independent given... A random variable is a term used in mathematics and statistics. ... This article defines some terms which characterize probability distributions of two or more variables. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ...

The probability that the sun will rise tomorrow

Let p be the long-run frequency of sunrises, i.e., the sun rises on 100 × p% of days. Prior to knowing of any sunrises, one is completely ignorant of the value of p. Laplace represented this prior ignorance by means of a uniform probability distribution on p. Thus the probability that p is between 20% and 50% is just 30%. This must not be interpreted to mean that in 30% of all cases, p is between 20% and 50%; that would be a frequentist philosophy of applied probability. Rather, it means that one's state of knowledge (or ignorance) justifies one in being 30% sure that the sun rises between 20% of the time and 50% of the time -- that is a Bayesian philosophy of applied probability. Given the value of p, and no other information relevant to the question of whether the sun will rise tomorrow, the probability that the sun will rise tomorrow is p. But we are not "given the value of p". What we are given is the observed data: the sun has risen every day on record. Laplace inferred the number of days by saying that the universe was created about 6000 years ago, based on a literal construction of the Bible. To find the conditional probability distribution of p given the data, one uses Bayes theorem, which some call the Bayes-Laplace rule. Having found the conditional probability distribution of p given the data, one may then calculate the conditional probability, given the data, that the sun will rise tomorrow. That conditional probability is given by the rule of succession. The probability that the sun will rise tomorrow increases with the number of days on which the sun has risen so far and would decrease as the number of days on which the sun has failed to rise increases. In mathematics, the uniform distributions are simple probability distributions. ... Statistical regularity has motivated the development of the relative frequency concept of probability. ... Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of statements, or to the degree of belief of rational agents in the truth of statements; when used with Bayes theorem, it then becomes Bayesian inference. ... The Bible (Hebrew: תנ״ך tanakh, Greek: η Βίβλος hÄ“ biblos) (sometimes The Holy Bible, The Book, Word of God, The Word Scripture, Scripture), from Greek (τα) βίβλια, (ta) biblia, (the) books, is the name used by Jews and Christians for their (differing but overlapping) canons of sacred texts. ... This article defines some terms which characterize probability distributions of two or more variables. ... Bayes theorem is a result in probability theory, which gives the conditional probability distribution of a random variable A given B in terms of the conditional probability distribution of variable B given A and the marginal probability distribution of A alone. ...

Mathematical details

The proportion p is treated as a uniformly distributed random variable. (Some who take an extreme Bayesian approach to applied probability insist that the word random should be banished altogether from probability theory, on the grounds of examples like this one. This proportion is not random, but uncertain. We assign a probability distribution to p to express our uncertainty, not to attribute randomness to p.)

Let X_i be the number of "successes" on the ith trial, 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. In the theory of probability and statistics, a Bernoulli trial is an experiment whose outcome is random and can be either of two possible outcomes, called success and failure. ... In probability theory and statistics, the Bernoulli distribution, named after Swiss scientist James Bernoulli, is a discrete probability distribution, which takes value 1 with success probability and value 0 with failure probability . ... In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Two random variables X and Y are conditionally independent given...

Bayes' theorem says that in order to get the conditional probability distribution of p given the data X_i, i = 1, ..., n, one multiplies the "prior" (i.e., marginal) probability measure assigned to p by the likelihood function Bayes theorem (also known as Bayes rule) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ... In statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: and also any other function proportional to such a function. ...

where s = x₁ + ... + x_n is the number of "successes" and n is of course the number of trials, and then normalizes, to get the "posterior" (i.e., conditional on the data) probability distribution of p. (We are using capital X to denote a random variable and lower-case x either as the dummy in the definition of a function or as the data actually observed.) The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics. ... In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...

The prior probability density function is equal to 1 for 0 < p < 1 and equal to 0 for p < 0 or p > 1. To get the normalizing constant, we find In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

(see beta function for more on integrals of this form). A separate article treats the beta-function (written with a hyphen) of physics. ...

The posterior probability density function is therefore

This is a beta distribution with expected value In probability theory and statistics, the beta distribution is a continuous probability distribution with the probability density function (pdf) defined on the interval [0, 1]: where α and β are parameters that must be greater than zero and B is the beta function. ... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical...

Since the conditional probability of tomorrow's sunrise, given the value of p, is just p, the law of total probability tell us that the probability of tomorrow's sunrise is just the expected value of p. Since all of this is conditional on the observed data X_i for i = 1, ..., n, we have Nomenclature in probability theory is not wholly standard. ...

Thus if the sun has risen every morning for 2,000,000 consecutive mornings, and no other data are available, Laplace would have us conclude that the probability of tomorrow's sunrise is

The probability that the sun will not rise tomorrow would then be slightly less than one in two million.