Fair Decisions With Random Generators

Lesson 2 of 2: Mapping, the Remainder, and Rejection

In this lesson:

• Map a generator's range into equal blocks
• Spot the bias when the count doesn't divide evenly
• Fix it with rejection sampling and verify

What You Will Be Able to Do

By the end of this lesson, you should be able to:

1. Map a random generator's range into equal blocks for choices
2. Detect the bias when the count doesn't divide the range evenly
3. Fix it with a divisible range or rejection sampling, verifying each is

From Dice to a 1-to-100 Generator

A die has 6 equally likely faces. A generator gives equally likely integers — say 1 to 100.

Same idea: map them into equal blocks.

For k Choices, Generate 1 to k

The cleanest approach: generate an integer from 1 to k directly.

Each value is equally likely, so each choice gets .

Map a Larger Range Into Equal Blocks

For 2 choices on a 1-100 generator: 1-50 and 51-100, each .

A Fair Generator, an Unfair Mapping

The generator can be perfectly fair while the mapping is biased.

Two separate questions: is the generator fair? Are the blocks equal?

Predict: Is This Split Fair?

A 1-100 generator, 3 choices, split 1-33 / 34-66 / 67-100.

Is each choice equally likely? Commit before advancing.

Count the Blocks: 33 / 33 / 34

The third choice gets — more than the others' .

Why: 100 Is Not Divisible by 3

remainder .

That leftover 1 has to go somewhere — and wherever it goes gets favored.

Fix 1: Use a Divisible Range

Generate 1 to 99 instead of 1 to 100.

Now split into 1-33 / 34-66 / 67-99 — each exactly .

Fix 2: Reject and Redraw (Rejection Sampling)

Keep 1-100, but reject 100 and redraw. Use only 1-99, split evenly.

Generalize: Reject the Top (n mod k)

To choose among k options with n equally likely outcomes:

Reject the top outcomes, split the rest into equal blocks, redraw if needed.

Verify Each Choice Is Exactly 1/k

A procedure is fair only if you can show each choice's probability is .

After any fix, count the block and confirm the fraction.

Resolve the Teaser: 5 With a Die

Five choices, a 6-sided die. .

Use faces 1-5 for the five choices; re-roll on 6.

Your Turn: Find and Fix the Bias

A 1-1000 generator picks among 7 prizes, split into blocks of about 143.

Is it fair? If not, fix it and verify each is .

What You Learned This Lesson

✓ Map a range into equal blocks — generate 1 to k when you can

✓ A leftover remainder biases one choice

✓ Reject the leftovers, redraw, and verify

Where Fair Randomness Goes Next

Fair random assignment underlies randomized sampling and experiments (HSS.IC).

And rejection sampling is how computers generate unbiased random integers.