Using Simulation to Reject or Fail to Reject a Claim

Start with a claim worth testing, like a coin is fair. You can't rerun history to check it, so you build a model of pure chance instead: flip a virtual fair coin 100 times, record the number of heads, and repeat that thousands of times. That gives a distribution of outcomes chance alone would produce. Most results cluster in the middle, with fewer and fewer as you move toward the extremes.

Once you have that distribution, mark off the middle chunk that covers most of the simulated outcomes, often the middle 95 percent. That middle chunk is the range of results that wouldn't raise eyebrows if the claim were true. Then you locate your one real result on that same scale. If it falls inside the range, the data looks like ordinary chance variation. If it falls outside, far from where chance usually lands, you reject the claim: chance alone is an unconvincing explanation for what happened.

This connects directly to margin of error and conditional probability questions nearby in the unit: the simulated range is doing the same job a margin of error does, marking off plausible chance variation around an expected value. The catch is what happens when the real result lands inside the range. That does not prove the claim true. It only means the data failed to catch the claim in a lie. A biased coin could still produce a result that looks ordinary by luck.