Does the problem prove that tomorrow's sunrise is uncertain in an everyday practical sense?

It illustrates inductive modeling, not a practical astronomical assessment.

Why add one and two in the formula?

They follow from the specified uniform beta prior and Bernoulli likelihood.

AI Glossary / Statistics and probability / Intermediate

Sunrise problem

Definition

The sunrise problem asks how observing an event repeatedly should affect belief that it will happen again. Its traditional illustration is predicting another sunrise after many previous sunrises. It is a problem about induction and probabilistic assumptions, often associated with Laplace's rule of succession, rather than a modern method for forecasting astronomical events.

What it means

Repeated success provides evidence, but evidence does not automatically imply certainty. A model must connect past observations to the next outcome. The sunrise illustration makes those usually hidden assumptions visible: what stays constant, what prior beliefs are appropriate, and what other knowledge should be used?
A simple Bayesian model gives a clear answer under specified assumptions. That answer is conditional on the model. Counting past sunrises while ignoring physical knowledge is not an adequate scientific description of the actual astronomical question.

How it works

Represent each observed event as a success or failure. In the basic construction, outcomes are independent conditional on a shared, fixed success probability p. This conditional independence assumption allows observations to update uncertainty about p.
Assign a uniform prior distribution to p between zero and one. This is a Beta(1,1) prior. Uniformity is a modeling choice, not a consequence of having no knowledge under every possible parameterization.
After observing s successes and f failures, the posterior is Beta(s+1,f+1). The posterior predictive probability of success on the next trial is (s+1)/(s+f+2).
If all n observations are successes, the result becomes (n+1)/(n+2). This is Laplace's rule of succession in the familiar all-success setting. It approaches one as the successful record grows, but remains below one for every finite n.
A different prior produces a different result. With a Beta(a,b) prior, the next-success probability is (s+a)/(s+f+a+b). The data and prior both enter the prediction; changing the prior is not merely a formatting change.
Distinguish predicting the next success from predicting several successes or a universal law. Future observations share uncertainty about p, so their joint predictive probability is not generally the product of repeated copies of the current one-step probability.
Under a continuous beta prior, the exact parameter value p=1 has probability zero, even after finitely many successes. A model intended to assign positive probability to a perfectly reliable law needs a different prior structure, such as an explicit point mass at one.
Consider whether the stable-probability model is appropriate. Changing equipment, environmental conditions, user behavior, and dependence can undermine an extrapolation from past success. Mechanistic knowledge and evidence about changing conditions should inform the model.
The lesson is therefore broader than one formula: predictive confidence depends on how observations are connected to future events and on which possibilities the model allows.

Three practical examples

Reliability discussions: explain why observing no failures does not prove perfect reliability.
AI evaluation: illustrate smoothing of an estimated success probability after a small all-success test.
Forecasting education: distinguish repeated observations from a justified model of why a pattern should continue.

Think of it this way

Think of a machine that has passed every inspection so far. Its record is encouraging, but a forecast also depends on whether tomorrow's operating conditions resemble yesterday's. A count of successful inspections alone cannot describe every reason the next run might differ.

Worked example

Consider an original toy process with 12 observed successes and no failures. Assume conditionally independent trials with a fixed success probability and a Beta(1,1) prior. The posterior is Beta(13,1).
The predictive probability of the next success is 13/14, approximately 92.86%. If that next trial succeeds, the updated probability for the following success is 14/15. Therefore the probability that both next trials succeed is (13/14)(14/15) = 13/15, approximately 86.67%.
Simply squaring 13/14 would give approximately 86.22%, which is different. The two future outcomes are dependent in the predictive distribution because they share uncertainty about the same p, even though they are independent conditional on p.
For comparison, a Beta(2,2) prior would produce Beta(14,2) after the same observations and a one-step predictive probability of 14/16 = 87.5%. These are model-based illustrations, not estimates of the probability of a real sunrise.

Illustrative case study

Fictional case: Harbor Automation celebrates 12 successful demonstrations of a new workflow. A manager proposes describing it as failure-proof. The analyst uses the sunrise problem to show why an uninterrupted success record does not establish certainty. The team also notices that all demonstrations used the same easy inputs, so even the simple independent-trial model is doubtful. It broadens testing and reports the operating conditions alongside results. The formula helps explain assumptions; it does not replace representative evaluation.

Common misconceptions and mistakes

Do not present the rule of succession as an objective astronomical probability. A uniform prior is not universally neutral. Twelve demonstrations need not be 12 independent, representative trials. A high probability for the next success is different from certainty that every future trial succeeds. Changing the underlying process can invalidate an extrapolation even after a long successful record.

Business use

Use this concept to challenge overconfident claims based on zero observed failures and to explain Bayesian smoothing. In operational decisions, combine observed performance with domain knowledge, realistic test coverage, and explicit assumptions about stability. Avoid treating an illustrative probability as a guarantee or a substitute for engineering evidence.

Check your understanding

What is the one-step prediction after 12 successes under the uniform prior?

Answer: 13/14.

Why is the two-success probability not simply its square?

Answer: Future outcomes share parameter uncertainty.

Would another prior necessarily give the same answer?

Answer: No.

Related terms

Bayesian inference
Posterior predictive distribution
Induction
Conditional independence
Rule of succession
beta distribution
exchangeability
additive smoothing

Learn next

Study beta-binomial updating, then compare parameter uncertainty with uncertainty about the model itself. Explore how point-mass priors represent hypotheses of exact perfection or universal regularity.

Sources

Wagenmakers, Zabell and Gronau, J. B. S. Haldane's Rule of Succession: https://arxiv.org/abs/2307.09489 Ollivier, Laplace's rule of succession in information geometry: https://arxiv.org/abs/1503.04304 The 12-success example and prior comparison are original.

Previous: Sunk Cost Fallacy Bias Next: Super column