Coin Toss
No single toss can be predicted. Toss it enough times, though, and the share of tosses that land heads settles on the coin’s true odds. This is the law of large numbers, and it is why casinos and insurers can plan around outcomes that are random one at a time. Load the coin and the share settles on a different value, just as surely. Streaks are part of the picture too: long runs of heads or tails turn up more often than most people expect.
What is the law of large numbers?
The law of large numbers says that the average of many independent trials converges on the expected value. For a coin, the share of heads after n tosses gets closer to p, the chance of heads on a single toss, as n grows. Jacob Bernoulli proved this in Ars Conjectandi, published in 1713. The coin does not correct itself along the way. An early run of heads is never balanced out by extra tails; it is diluted by the many tosses that follow.
What does the shaded band show?
At any number of tosses n, the band covers the range p ± 1.96√(p(1−p)/n). The share of heads lands inside it 95% of the time. For a fair coin the band is ±0.31 wide after 10 tosses, ±0.10 after 100 and ±0.03 after 1,000. Because the width shrinks with √n, halving it takes four times as many tosses. Over a long run the line will still step outside the band now and then, since 95% is not 100%.
Why is the x-axis on a log scale?
Most of the movement happens early, when each toss shifts the share a lot. On an ordinary axis, the first 100 tosses of a 10,000-toss run would be squeezed into the left 1% of the chart. A log scale gives each tenfold step (1–10, 10–100, 100–1,000) the same width, so you can see the share settle at every stage.
Are long streaks a sign something is wrong?
No. In 100 tosses of a fair coin there is about an 80% chance of a run of six or more identical faces somewhere. The longest run grows roughly with log2(n), so streaks get longer as you keep tossing. Each toss is still independent: after five heads in a row, the next toss is heads with probability p, as always. Expecting a correction is the gambler’s fallacy.