The Mathematics of Random Splits

Splitting a quantity into random proportions is one of the oldest problems in probability. Imagine snapping a stick at random points and measuring the pieces. Mathematicians call the resulting partition a random composition, and every valid split of 100 into 2 positive integer parts carries exactly the same probability. The algorithm behind this tool places 1 unique break point among 99 possible positions, sorts them, and measures the gaps. Clean, fast, and provably fair.

The Broken Stick Surprise

Ask someone to guess the average largest piece when breaking a stick into 2 equal-probability parts. Most people guess something close to 50%. The real answer is far higher. For 2 parts, the expected largest piece follows the harmonic number formula: H(2)/2 of the total. That works out to roughly 75%. The golden marker labeled E[max] on the distribution line above tracks this theoretical value. Keep splitting and watch your dots cluster around it.

For two parts specifically, the expected largest piece is approximately 75%. Three out of four times, one side gets more than half. That asymmetry is genuine randomness at work, and it explains why real-world resource distributions so rarely land at 50/50 even when the underlying process is perfectly fair.

Uniformity on the Simplex

Geometrically, all possible ways to split 100 into 2 non-negative parts live on a shape called the (1)-simplex. For two parts, the simplex is a line segment. For three parts, it is a triangle. For four, a tetrahedron. The random composition algorithm samples uniformly across this geometric object, meaning every point on the surface has equal probability. Mathematicians Peter Gustav Lejeune Dirichlet and later George Pólya formalized the distribution of random partitions, and the Dirichlet distribution with all parameters equal to 1 describes exactly the continuous analog of what this tool produces with integer rounding.

Fairness and Entropy

Each split calls crypto.getRandomValues(), the Web Cryptography API built into every modern browser. This function draws from hardware-level entropy sources: thermal noise, electrical jitter, and other physical processes in your device. The random break points are selected through rejection sampling from these hardware seeds, ensuring uniform coverage of all valid positions. Your browser does all the computation. The server delivers this page and nothing more. It never knows your result.

In the Classroom

Random proportions are natural teaching tools for fractions, percentages, and resource allocation. Give students a scenario: a school has a budget of 100 units to split among 2 programs. Have each student visit /ratio/2 and split repeatedly. Record the largest allocation each time. After 20 splits per student, compute the class average of the largest part. It converges toward the harmonic number prediction, demonstrating how probability theory predicts aggregate behavior even when individual results vary wildly.

For a pie chart exercise, have students generate a split with /ratio/5, then draw the corresponding pie chart by hand. Translating percentages into angles (each 1% equals 3.6 degrees) reinforces the connection between fractions, percentages, and geometric representation. The tool requires no accounts, collects no student data, and sets no cookies.

Private by Architecture

Every split happens inside your browser. The server delivers the page. Your device generates the proportions. Results live in your browser's memory and localStorage, fully under your control. Sharing this URL sends the tool configuration to another person. Their device generates its own independent split from its own entropy source. Two people visiting the same link produce completely different results every time.