How does the two-coloured-balls (interactive ZK) example work, and why does repetition matter?

Alice is colour-blind and doesn't trust Bob's claim he can see colour. She uses two visually identical balls — one red, one blue — and an interactive game where she challenges Bob to consistently identify when she's swapped them or not.

The protocol:

1. Alice holds a red ball in one hand, blue in the other (visually identical to her).
2. She shows Bob both, then puts her hands behind her back.
3. She either swaps the balls between hands or doesn't — Alice's secret choice, hidden from Bob.
4. She brings her hands back out and asks Bob: "Did I swap or not?"
5. Bob answers.

The reasoning:

• If Bob can see colour, he always answers correctly (because he sees which colour is in which hand).
• If Bob is lying (can't see colour), he must guess with 50/50 chance — gets it right half the time on average.

Why repetition is necessary: one correct answer proves nothing — Bob could have guessed. After n rounds, the chance Bob guessed every one correctly without actually seeing colour is 1/2ⁿ.

After ~20-30 rounds, Alice is convinced beyond reasonable doubt.

Why this is "zero-knowledge": Alice never learns anything about how Bob distinguishes colours. She only learns the fact "Bob can distinguish them."

Important phrasing:

"Zero-Knowledge-Beweise basieren auf statistischen Garantien dafür, dass Alice nichts lernt."

ZK proofs are statistical, not absolute. With enough rounds the probability of error becomes negligible.

Tip: This is the template for almost every interactive ZK protocol — the prover commits, the verifier challenges, the prover reveals consistently if and only if they know the secret, and repetition drives the cheating probability to zero exponentially.