Experiments you can run yourself.
Switching doors after a goat is revealed gives a 2/3 chance of winning, not 1/2.
With just 23 people the chance of a shared birthday exceeds 50%.
As the number of random points grows, the estimate of π becomes more and more accurate.
Summing many independent random events yields a bell-shaped (normal) distribution.
As the number of needle tosses grows, the estimate of π becomes more and more accurate.
In the Nash equilibrium of the prisoner's dilemma, both players defect.
The probability of a random point landing in a smaller region equals the ratio of areas.
A result that holds in every group can reverse once the data are combined.
A positive result from an accurate test for a very rare disease most often means that you are healthy.
In natural data the leading digit is more often 1 (~30%) than 9 (~5%).
It is impossible to design a perfectly fair voting system.
The optimal strategy is to reject about the first 37% (1/e) of candidates, then pick the first one who is better.
A clever cycle-following strategy gives the prisoners a ~31% chance instead of practically zero.
Out of the chaos of randomly bouncing balls a bell curve (the normal distribution) emerges.
After five reds in a row the chance of black is still ~½ — the roulette wheel has no memory.