Rule of Succession Formula

See: Cromwell's Rule, Binomial Distribution, Sunrise Problem, Conditionally Independent Random Variables.

References

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/rule_of_succession Retrieved:2015-6-15.
- 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. ^[1]
  The formula is still used, particularly to estimate underlying probabilities when there are few observations, or for events that have not been observed to occur at all in (finite) sample data. Assigning events a zero probability contravenes Cromwell's rule, which can never be strictly justified in physical situations, albeit sometimes must be assumed in practice.

↑ Laplace, Pierre-Simon (1814). Essai philosophique sur les probabilités. Paris: Courcier.

(Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/rule_of_succession#Statement_of_the_rule_of_succession Retrieved:2015-6-15.
- If we repeat an experiment that we know can result in a success or failure, n times independently, and get s successes, then what is the probability that the next repetition will succeed?
  More abstractly: If X₁, ..., Xn+1 are conditionally independent random variables that each can assume the value 0 or 1, then, if we know nothing more about them, : [math]\displaystyle{ P(X_{n+1}=1 \mid X_1+\cdots+X_n=s)={s+1 \over n+2}. }[/math]