One of my favorite math activities, albeit one that requires a bit of patience, involves flipping a coin 100 times. On one side of a sheet of paper, you record the actual results. On the other side of the paper, you make up the results. The final product is a combination of 200 heads and tails (hence the patience). You then pose a question of someone who didn’t observe the process: which side holds the actual results?
The answer is pretty simple: almost always, it’s the side with the longest streak of consecutive heads or tails. When we make up the results, we are reticent to record (say) eight straight heads or tails; it seems improbable.