Random Numbers Without Bias, Explained

After reading this you will know how to draw fair random integers, why the common random() % range trick quietly favours small numbers, and how to read a stream of draws without fooling yourself about what "random" means.

What a uniform random integer actually is

A uniform random integer generator picks a whole number from a fixed range so that every value has the same chance. If you draw from 1 to 6, each of the six faces should come up with probability 1/6 \approx 0.1667. Nothing is "hot" or "due". A fair die that just landed on 6 nine times in a row still has probability 1/6 of landing on 6 again.

Here is the hook. Roll a fair die 60 times and you expect about 10 of each face, but you will almost never get exactly 10 of each. A spread of 6 to 14 per face is completely normal. People read those wobbles as patterns. They are not patterns. They are the ordinary noise of small samples, and the math below tells you exactly how big that noise should be.

When to use it, and when not

Reach for a plain random integer when every outcome deserves equal odds and you want a value or a set of values: pick a winner from ticket numbers 1 to 5000, roll dice, generate a PIN, sample rows from a spreadsheet, or draw six lottery balls that must not repeat.

Do not use it when the outcomes are not equally likely. If one raffle entry bought ten tickets and another bought one, you want the ten-ticket entry to win ten times as often. That is a weighting problem, not a uniform draw. Do not use raw integers to assign people to teams either, because keeping team sizes equal is a shuffle, not independent picks.

The bias in the naive method, and how rejection sampling fixes it

Suppose your random source gives you an integer from 0 to 15 (16 equally likely values) and you want a number from 0 to 9. The tempting one-liner is:

Here R is the raw value from 0 to 15, and \bmod is the remainder after division. Count how many raw values map to each result. Values 0 and 10 both give 0. Values 1 and 11 both give 1. This continues up to 5 and 15, which both give 5. But values 6, 7, 8, 9 have no partner, so results 6 through 9 come from one raw value each.

So results 0 to 5 each have probability 2/16 = 0.125, while results 6 to 9 each have probability 1/16 = 0.0625. The small numbers are exactly twice as likely. That is real bias, and it never averages out no matter how many draws you take.

Rejection sampling removes it. Find the largest multiple of your range size that fits inside the raw range, then throw away and redraw any raw value at or above it.

N is the number of raw values available (16 here), n is your range size (10), and the brackets mean round down. So \text{limit} = \lfloor 16/10 \rfloor \cdot 10 = 10. You accept raw values 0 to 9 and reject 10 to 15. Every accepted value maps to exactly one result, so every result has probability 1/10 = 0.1000. The cost is that you sometimes redraw. In this example you reject 6 of 16 raw values, so about 37.5% of the time you draw again. In practice the raw range is enormous (32 bits, over 4 billion values), so rejections are vanishingly rare and every value in your range is exactly equally likely.

A worked example: reproducing the demo

Reading a stream of draws

The single most useful number for judging fairness is the standard deviation of counts. Roll a fair n-sided die m times. The count for any one face follows a binomial distribution with expected value and standard deviation:

\mu is the average count you expect for each face, and \sigma measures the typical wobble around it. For 60 rolls of a 6-sided die, \mu = 10 and \sigma = \sqrt{60 \cdot (1/6) \cdot (5/6)} \approx 2.89. So a face landing 7 or 13 times is about one standard deviation off, entirely ordinary. A face landing 20 times is (20 - 10)/2.89 \approx 3.46 deviations out, which is genuinely rare and worth a second look.

Watch the sample size shrink the noise

The wobble does not grow with more rolls, it grows slower than the total. The relative spread \sigma / \mu shrinks as you roll more, which is why long runs look flat and short runs look lumpy.

Common mistakes

Believing in "due" numbers

Each draw is independent. After five heads, the next flip is still 50/50. A number that has not appeared in a lottery for months is no more likely next time. The technical name for the opposite belief is the gambler's fallacy.

Confusing unique draws with independent draws

Independent draws can repeat. If you need distinct values (six lottery balls, five raffle winners from a list), use the no-repeat mode. If you want each draw fresh and repeats allowed (dice, coin flips), use the with-repeat mode. They answer different questions.

Trusting an off-the-shelf Math.random() for anything sensitive

Ordinary language-level random functions are fast but predictable: an attacker who sees enough outputs can reconstruct the internal state and forecast future values. This tool draws from the browser's cryptographic source instead, so the outputs are not predictable from past outputs. Use cryptographic randomness for tokens, passwords, and anything an adversary could profit from guessing.

Reading a small run as biased

Before you accuse a generator of bias, compute \sigma. A face two or three counts off the mean in a 60-roll run is expected, not suspicious.

Related tools

When outcomes are not equally likely, the Weighted Random Picker lets some entries earn better odds while staying fair to the weights you set. To split names into balanced groups instead of drawing values, use the Team & Pair Randomizer, which shuffles rather than samples so group sizes stay even. For competitions, the Tournament Bracket Generator builds single-elimination brackets with byes, seeded or fully random.

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