Law of large numbers
Flip a fair coin or roll a fair die, one at a time and then in bursts. The line is the share of successes so far, a head or the face you chose; the dashed line is its probability. Early on the share jumps about; as the trials pile up it settles.
Goes with chapter 1. Probability and events
Set up
Number of flips
Where this run ends: 1,000 flips, 536 heads, a share of 0.5360, which is 0.036 away from p = 0.5.
What to look for
With a few trials the share can be far from p: three heads in the first four flips is a share of 0.75. That is not the coin being unfair; it is too few flips to tell. Run 10,000 and the share will very probably end within a percentage point of 0.5.
The law does not say that a run of heads makes tails more likely. Every flip is still 50:50. The early surplus is not paid back; it is diluted, because it stays the same size while the number of flips grows.
Try the same run length with a new run a few times: the line wanders differently each time, but it ends close to p, and usually closer the longer the run.