Mathematics
Long-Term Average
Quick fact
Even after 10,000 coin flips, the difference between heads and tails can be hundreds, yet the ratio of heads to total flips will be remarkably close to 0.5.
Why this is interesting
When you flip a coin just a few times, the results can be wildly uneven—but why does the average always creep toward 50% heads if you keep flipping?
Read the full explanation
Understanding Long-Term Average
Imagine betting on a fair coin: heads you win $1, tails you lose $1. After one flip, you're either up or down by $1. After ten flips, your average gain per flip could be anywhere between -$1 and $1. But after a million flips, your average gain per flip will almost certainly be very close to $0. This is the long-term average: the number you'd expect to see if you could perform an infinite number of trials. It's the same idea behind rolling dice, measuring rainfall, or any repeated random event—the average stabilizes as more data accumulates, revealing the underlying true mean.
A deeper explanation
The long-term average works because of the law of large numbers, a theorem stating that as the number of independent trials increases, the sample average converges to the expected value. This convergence is not about canceling out previous deviations but about the growing denominator dwarfing any imbalance. For example, after 100 flips you might have 60 heads (0.6), but after 10,000 flips, even a 100-flip imbalance becomes just 1% of the total. The mechanism relies on independent repetitions and finite variance. Without independence (like in weather patterns) the concept still applies but requires care. Understanding the long-term average is crucial for insurance premiums (predicting average claims), gambling (house edge), and scientific experiments (estimating true effects). It reassures us that while short-term chaos dominates, order emerges in the long run.