Random Number Generator

Definition

Let min, max ∈ ℤ with min ≤ max. The range length is n = max − min + 1. The support set is S = {min, min + 1, …, max} = [min, max] ∩ ℤ. The discrete uniform distribution on S assigns each s ∈ S the probability P(X = s) = 1/n. This calculator draws one or more independent samples from S (with replacement) or samples without replacement when no repeats is checked.

For a single draw with minimum 1 and maximum 6 we have n = 6 and P(X = k) = 1/6 for k = 1, …, 6. That is the classic fair-die model on a finite integer set — not a continuous uniform on a real interval.

The outcome is random in the sense of the generator implemented in the browser, not in the sense of a paper proof about an infinite i.i.d. sequence. When bounds are extreme, check the integer domain first, then the ±10⁹ limit and the count limit of 100.

Historical and conceptual context

Donald E. Knuth in The Art of Computer Programming, Volume 2 (Seminumerical Algorithms), treats computer “random” numbers carefully: deterministic PRNG sequences versus ideal randomness, tests, and pitfalls when carrying bits onto a finite range. Modern browsers also expose a CSPRNG through the Web Crypto API (crypto.getRandomValues), documented on MDN and in the W3C specification.

NIST publishes guidance for deterministic generators and entropy sources in the SP 800-90 series (including 800-90A for DRBGs). This page does not claim NIST certification: it uses the browser API and honestly separates CSPRNG from Math.random. NIST documents provide conceptual context (what cryptography demands), not a quality seal for this calculator.

Conceptually separate: (1) the mathematical model — discrete U(S); (2) the bit source — CSPRNG or PRNG; (3) the map from bits onto S — here rejection sampling to avoid modulo bias (unequal probabilities). Fortune wheels, lotteries, and official draws are separate legal and hardware procedures.

Formal properties

For X uniform on S: E[X] = (min + max)/2, Var(X) = (n² − 1)/12. The mass function is constant on S and zero outside S. With m independent draws with replacement, (X₁, …, Xₘ) has a product distribution; repeated values are allowed.

With m ≤ n and sampling without replacement, the result is a uniformly chosen ordered m-tuple from S (draw order), equivalent to a partial permutation. The number of ordered m-tuples without repeats is P(n, m) = n!/(n − m)!. When m > n the event is impossible — the calculator reports an error instead of inventing a result.

The identity n = max − min + 1 is exact in integer arithmetic within the ±10⁹ limit. Shifting the range (min + c, max + c) leaves n and uniformity on the new set unchanged.

Full derivation

Step 1. Read integers min, max and count m ∈ {1, …, 100}. Reject fractions and min > max.

Step 2. Compute n = max − min + 1. For one draw choose U ∈ {0, …, n − 1} uniformly, then return min + U.

Step 3 (CSPRNG). Read a 32-bit value from crypto.getRandomValues. Let M = 2³² and L = ⌊M / n⌋ · n. Reject samples ≥ L, then take the remainder modulo n. Each residue 0…n−1 then has equal probability (no modulo bias when n does not divide M).

Step 4 (fallback generator). When Web Crypto is unavailable: U = ⌊Math.random() · n⌋. ECMAScript does not guarantee cryptographic quality for Math.random.

Step 5 (no repeats). Build the pool [min, …, max] and run m steps of partial Fisher–Yates: at index i draw j ∈ {i, …, n − 1}, swap, append pool[i].

Step 6. Example: min = 1, max = 6, m = 1 → one of six values with P = 1/6. min = 1, max = 49, m = 6, no repeats → six distinct values from 1…49.

Special cases

min = max: n = 1, the only possible value is that shared bound; without repeats only m = 1 is allowed. Range −10…10: n = 21, zero is included. m = 100 on a small n with replacement is allowed; without repeats requires n ≥ 100.

An empty count field inserts m = 1. Values outside ±10⁹ or non-integer input produce an error instead of silent truncation.

The result is not a sequence tested by Diehard/TestU01 — it is a single educational draw in the browser.

Numerical notes

Math.random returns a Number in [0, 1) and is not intended for secrets (passwords, tokens, keys). Even with a CSPRNG, simply taking the remainder x % n introduces slight bias when n does not divide 2³² — hence rejection sampling in this calculator.

For very large n (order 10⁹) a no-repeats pool would consume too much memory; the limits m ≤ 100 and |min|,|max| ≤ 10⁹ keep the tool practical in a browser. The result panel shows n, P = 1/n, and the source label (Web Crypto or the fallback generator).

Do not use this page for digital signatures, legal lotteries, or production password generation. For cryptography use dedicated APIs and procedures aligned with current guidance (e.g. the NIST SP 800-90 context), not an interactive educational calculator.

Related concepts and sources

Related: combinations C(n, k) (subset counts), permutations, modulo (remainder, not a draw), continuous uniform (a different model). To count ways use the combinations page; here the result is concrete numbers.

Sources: Knuth, TAOCP vol. 2 — Seminumerical Algorithms (random numbers chapter); NIST SP 800-90A Rev. 1 (DRBG — context, not a certificate for this page); MDN — Crypto.getRandomValues; Wikipedia — Discrete uniform distribution; W3C Web Cryptography API.