There is the task on probability:
We have a fair coin and we intend to make 8 coin toss. After 4 coin tosses we have 3 Tails and 1 Head, what is the expected result after all 8 coin tosses.
Answer 5 Tails, 3 Heads, since one can expect those 4 coin tosses to be 50/50.
At the same time we know that for large numbers we need to have 50/50 probability, regardless of temporary overweight of one side over other. So Heads "should catch up" with Tails if we toss coin long enough.
But isn't there a contradiction: On one side we expect to have 50/50 result, on the other we admit that after some overweight we expect rest of the tosses to be 50/50, so this overweight should at least still be present.
The overweight gets smaller and smaller as the amount of tosses gets larger, so taking the limit to infinity this overweight disappears.
$$\frac{x_1+\ldots+x_N+X_{N+1}+\ldots+X_n}{n}=\frac{x_1+\ldots+x_N}{n}+\frac{X_{N+1}+\ldots+X_n}{n}\to 0.5$$ as $n\to\infty$, where we already now the first $N$ outcomes, because $\frac{x_1+\ldots+x_N}{n}\to 0$ and $\frac{X_{N+1}+\ldots+X_n}{n}\to 0.5$ as $n\to\infty$.