Why a coin’s proportion settles down
For a fair coin, each flip has a one-half chance of landing heads. The law of large numbers says that as the number of flips grows, the proportion of heads tends to get closer to one-half. It does not say that heads and tails must alternate, or that every batch will split evenly.
The mechanism is that each new flip adds only one result to a running total. Early flips can strongly sway the proportion: after one flip, the result is either 0% or 100% heads. Later, the same single flip is a much smaller part of the whole. A run of heads still counts, but it has less power to shift a large tally.
Closer on average, not guaranteed
For a fair coin, the typical size of the proportion’s deviation from one-half shrinks roughly in proportion to one over the square root of the number of flips. So multiplying the number of flips by 100 cuts that typical deviation by about a factor of 10—not by 100. This is why getting noticeably closer can require far more flipping than intuition suggests.
“ Tends to” matters: the law describes what happens with increasing sample size, not a deadline by which balance must arrive. A long sequence can end with a sizable streak, and no finite number of flips guarantees an exact 50–50 split. The practical change is subtler: as the tally grows, even a dramatic run becomes a smaller fraction of the record.
