A coin’s true bias is hidden. Flip it, weigh the evidence against a fair coin, and decide whether it is fair or rigged — then the truth is revealed. Judge many coins and watch how often a p-value rule is wrong.

Where is the data?

This sim is self-contained — the data is right here. Each coin you judge becomes one square on the Scoreboard, coloured by whether your call was right. Over many coins, the board tallies the false alarms (Type I) and misses (Type II).

How the numbers work

The pile shows 100 pretend fair coins, each flipped as many times as yours — the fraction at least as lopsided as your result is a simulated p-value. The reported value is the exact two-sided binomial probability under p = 0.5. The surprise meter maps that p-value to how unusual the result would be for a fair coin. On the Scoreboard, the “call it rigged when p < α” rule flags a share of truly fair coins that settles near α itself.

Standards

• CCSS HSS-IC.A.2 — decide whether a model is consistent with a data-generating process, using simulation (the standard’s own coin example).
• AP Statistics Unit 6 — significance test for a proportion, interpreting a p-value, and Type I / Type II error.

Design intent

• The coin’s true bias is hidden and then revealed, so inference becomes a decision that can be checked — the one thing a test run on a finished dataset cannot show.
• The p-value is shown as a countable pile of simulated fair coins before it is ever a number; error rates emerge on the Scoreboard rather than being asserted.

Discussion prompts

• Your coin passed as fair — does that prove it is fair? What would convince you it is biased?
• If we call a coin rigged whenever p < 0.05, how often will we wrongly accuse a perfectly fair coin? Check the Scoreboard.
• A subtle bias slips through more often than a strong one. Why does catching it take more flips?

How do I explore?

1. Press Flip for one toss, or Flip ×100 for a hundred, and watch the heads pile up. Press New coin any time for a fresh mystery coin.
2. Look at the pile of 100 pretend fair coins and the surprise meter: how unusual is your result if the coin were actually fair?
3. Make the call — Fair or Rigged — then see the coin’s true bias and whether you were right.

What should I do/notice?

• Even a fair coin wobbles — landing a little off from half the time is normal.
• The more you flip, the more the evidence settles. A small handful of flips can fool you.
• A calm p-value never proves the coin is fair — it only fails to catch a problem.

What about the data?

Each coin you judge becomes one square on the Scoreboard. Green means a correct call; the other colours are the two ways to be wrong. Over many coins, the board shows how often the rule is fooled.