Computer Science & Algorithms
Section 1
Core Idea
Laplace's rule (or rule of succession) estimates the probability of an event that has occurred k times in n trials as (k + 1) / (n + 2). It's a simple Bayesian prior: start from "one success and one failure" in your head, then update with observed data. When you have little data, it prevents 0% or 100% estimates and nudges you toward the middle. When you have lots of data, it converges to the observed frequency. Useful for rare events, A/B tests with small samples, and any setting where raw k/n would be overconfident.
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Section 2
How to See It
Understanding & Analyzing
You're seeing Laplace's Law when someone estimates a rate (e.g. conversion, defect rate) as (successes + 1) / (trials + 2) instead of raw successes/trials, especially when counts are small.
Deciding & Judging
You're seeing Laplace's Law when a decision uses a smoothed probability to avoid treating "0 out of 5" as 0% or "5 out of 5" as 100%, and instead uses a conservative prior that shrinks toward ½.
Section 3
How to Use It
When you have counts (k successes in n trials), use (k + 1) / (n + 2) as a simple Bayesian estimate. It's especially useful for small n: it avoids overconfidence from 0/n or n/n and gives a reasonable default when data is sparse. For large n, it barely differs from k/n.
Decision filter
"Do we have small counts and raw proportions that look too extreme (0% or 100%)? If yes, Laplace's rule gives a smoothed estimate that accounts for uncertainty and avoids overconfidence."
As a founder
Use Laplace smoothing when interpreting early conversion rates, support ticket rates, or any metric with few observations. "3 out of 10 signed up" → (3+1)/(10+2) = ⅓ instead of 30%. Stops you from overreacting to tiny samples and aligns with Bayesian updating when you have little data.
Section 5
Founders & Leaders
Ed ThorpMathematician; author, Beat the Dealer; quantitative investor
Thorp used careful probability estimates under uncertainty — in blackjack, in options, in markets. Laplace's rule is the same spirit: don't treat small-sample proportions as true rates. Add a prior (the +1, +2) and update with data. Founders can use it for conversion, churn, or defect rates when trials are limited; it reduces overconfidence and supports better decisions with sparse data.
Section 7
Connected Models
Reinforces
Bayes Theorem
Laplace's rule is a simple Bayesian estimate: a uniform prior (equivalent to one success and one failure) updated by observed counts. Same logic as Bayes; closed-form for this case.
Tension
Average Rule
The average rule uses observed averages; Laplace's rule shrinks estimates when n is small. The tension: with lots of data, they agree; with little data, Laplace avoids 0/1 extremes that the raw average can give.
Leads-to
Signal vs Noise
Small samples are noisy. Laplace's rule is one way to separate signal from noise: don't treat noisy proportions as exact. Smooth first, then interpret.
Section 8
One Key Quote
— Pierre-Simon Laplace, Théorie analytique des probabilités"When the probability of an event is unknown, we may suppose it equal to any value between 0 and 1." The rule of succession makes that concrete: (k+1)/(n+2) is the posterior mean under a uniform prior.
Section 11
Summary & Further Reading
Laplace's rule estimates a probability from k successes in n trials as (k + 1) / (n + 2). It's a simple Bayesian prior that prevents 0% or 100% from tiny samples and converges to k/n when n is large. Use it for small-sample rates and A/B interpretation.
01
Book
Covers Bayesian inference and the rule of succession in a clear, rigorous way.
02
Book
Applies Laplace and Bayesian thinking to real-world decisions and small-sample problems.
03
Book
On overconfidence and small samples; Laplace's rule as a corrective for intuitive misestimation.