All simulators
Chance experiments
Toss coins and roll dice thousands of times. Watch frequencies settle on the true chance.
Toss coins and roll dice thousands of times. Watch frequencies settle on the true chance.
Toss the coin to start.
Bar height = experimental fraction; number on top = count. Tap a bar to follow it on the line below.
Grey funnel: if the theory is right, the fraction almost always stays inside it once there have been a few hundred trials (rare outcomes slip out more often early on). It narrows as the trials grow.
Theoretical probability is worked out by thinking, before anything happens. When all outcomes are equally likely, P = favourable outcomes ÷ total outcomes. A coin has 2 equally likely faces, so P(heads) = 1/2.
Experimental probability (relative frequency) is measured: repeat the experiment and work out times it happened ÷ number of trials. It changes from run to run, and after only a few trials it can be far from the theory.
With more and more trials the fraction settles down near the theoretical probability: the law of large numbers. The coin has no memory. Tails don’t “catch up”; early luck simply gets diluted by thousands of new tosses. The grey funnel on the line chart shows where the fraction is almost always found if the theory is right.
Takeaway: theory says what to expect, data says what happened. With enough data you can test a theory, or catch a loaded die. AI models learn probabilities from data in just this way.
Toss (total so far: 0)
“Keep going” speeds up by itself and stops at 1,00,000 trials.
Experimental: Toss to collect some data.
Theoretical: 2 equally likely faces, 1 of them is this one → 1/2 = 0.5.
Theoretical probability is worked out by thinking, before anything happens. When all outcomes are equally likely, P = favourable outcomes ÷ total outcomes. A coin has 2 equally likely faces, so P(heads) = 1/2.
Experimental probability (relative frequency) is measured: repeat the experiment and work out times it happened ÷ number of trials. It changes from run to run, and after only a few trials it can be far from the theory.
With more and more trials the fraction settles down near the theoretical probability: the law of large numbers. The coin has no memory. Tails don’t “catch up”; early luck simply gets diluted by thousands of new tosses. The grey funnel on the line chart shows where the fraction is almost always found if the theory is right.
Takeaway: theory says what to expect, data says what happened. With enough data you can test a theory, or catch a loaded die. AI models learn probabilities from data in just this way.
Things to try