The long run
Toss a coin again and again. The fraction of heads jumps around wildly at first, then settles down near 1/2. That settling is what a probability means as a long-run frequency.
From the slides: Why should anyone care about chance? and Probability and what happens when you learn something.
Trials in each run
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Run 1
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Farthest run from the exact value
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Exact value
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What to notice
- Early on, different runs disagree a lot. After thousands of trials they all sit close to the exact value.
- The runs never stop wobbling; they wobble less. Tick show the typical range: the band shrinks like 1/√n, so a hundred times as many trials gives answers about ten times as precise.
- In the roulette mode the running average is your winnings per ₹1 bet. It settles at −1/37 ≈ −0.027: the green zero is the casino's whole business model.
- Making this settling precise is the law of large numbers, proved near the end of the course.