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The same small block on the end of a short bar and of a long bar Two horizontal bars start from the same upright line, one above the other. The upper bar is short and the lower bar is more than five times longer. A small block of exactly the same size sits just beyond the end of each bar, separated by a gap. On the short bar that block is most of the length. On the long bar it is a sliver.

Nothing Ever Catches Up

About 10 minutes

Part 2 left two facts sitting next to each other that look like they cannot both be true. Across 3,600 rolls, seven ended up 14 short of its prediction and stayed short. And yet its share came out at 16.3% against a predicted 16.7%, which is very nearly perfect.

If nobody paid the 14 back, how did the share get fixed?

The answer is that the share was never fixed. It was buried.

The obvious idea, and why it is wrong

Say a coin comes up heads 15 times in its first 20 flips. Ten more heads than tails.

Almost everybody's instinct is that the coin now owes some tails. Not straight away, perhaps, but eventually — because it has to come out even in the end, so a run of tails must be waiting somewhere to balance the books.

Ask what would have to be true for that to work. The coin would have to know it is 10 heads ahead. So something, somewhere, would have to be keeping count.

Look at the coin. It is a disc of metal. It has no dent that deepens with each head, no wheel that turns, no marks that rub off. Nothing about it is different after 20 flips than before them. The next flip meets exactly the same coin the first flip met, and the coin has no way to lean.

A coin has no memory, so it cannot owe you anything. Neither can a die, a roulette wheel, a lottery machine, or a shuffled pack. That is not a rule someone made up. There is simply no part in any of them where the debt could be stored.

So watch what actually happens

Keep flipping. Suppose from flip 21 onwards the coin behaves impeccably: exactly half heads, half tails, for as long as you like. The lead of 10 is never repaid, because nothing is repaying it.

A heads bar and a tails bar growing together, ten flips apart Two horizontal bars start from the same upright line. The upper bar counts heads and the lower one counts tails. The heads bar is made of two pieces: a pale piece exactly as long as the tails bar, and a solid block on its end that is always ten flips wide. As the slider adds flips both bars grow, the solid block stays exactly the same size, and it becomes a smaller and smaller share of the whole bar. heads tails the solid block is the ten-flip lead flips so far: 80

The top bar is heads, the bottom bar is tails. Drag to keep flipping, and watch the gap between them.

The gap never changes. It is exactly ten flips wide at 20 and exactly ten flips wide at 200, because nothing has paid anything back.

What changes is everything around it. At 20 flips ten is half of everything on the page. At 200 flips the same ten is a sliver. The lead has not shrunk. It has been swamped.

Put numbers on the four moments:

Flips Heads Tails Heads' share
20 15 5 75%
50 30 20 60%
100 55 45 55%
200 105 95 52.5%

Keep going and it keeps falling: 50.5% at a thousand flips, 50.25% at two thousand. It never quite reaches half and it never needs to. The lead of 10 is still sitting there, whole and unpaid, at every single one of those rows.

That is how the share gets fixed: not by correction, but by dilution.

The part that sounds like a mistake

Real coins do not behave impeccably from flip 21, of course. So what does the gap really do?

I flipped 4,000 imaginary coins for each of four lengths and measured how far apart heads and tails typically ended up:

Flips Typical gap between heads and tails That gap as a share
20 about 4 17.6%
100 about 8 8.0%
1,000 about 25 2.5%
10,000 about 81 0.8%

Read the middle column downwards. The gap does not shrink towards zero. It grows. By ten thousand flips heads and tails are typically about eighty apart, twenty times further apart than they were at twenty flips.

Now read the right-hand column. The share of that same growing gap falls, all the way down, without pausing.

Both columns are describing one run. The coin drifts further and further from dead level in absolute terms, and closer and closer to dead level in percentage terms, and there is no contradiction, because they are two different measurements. Dividing by a total that is growing much faster wins.

A roulette wheel has come up red eight times running. A player bets everything on black, because "black is due". What is wrong with the reasoning?

Two things that are easy to swap

It is worth being exact about what is being denied here, because the true statement and the false one sound almost identical.

The second one is about a run you have not seen yet. The first is about one you already have, and once you have it, the coin has moved on and does not care.

Which raises a question the last three chapters have been quietly walking towards. If nothing ever corrects a run, then runs are not rare freaks that get smoothed away — they must just happen, and go on happening.

So how often? Should five heads in a row surprise you at all?