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A gappy tally on the left and a settled hill on the right Two bar charts stand on one table line. The chart on the left has only five bars at wildly different heights with empty gaps where nothing came up, the way a short run of dice rolls looks. The chart on the right has nine bars rising evenly to a peak in the middle and falling away again, the way a long run looks. A straight arrow between them points from the gappy chart to the smooth one.

How Many Rolls Before You Can Trust It

About 15 minutes

Counting the thirty-six rolls gave a clean answer: a hill, with seven on top at 16.7% and the two ends down at 2.8%. Twelve real rolls then produced no sevens at all.

Neither of those is a mistake. They are answers to two different questions, and the gap between them closes at a rate you can watch.

One run, written down as it happened

I rolled two dice 3,600 times and kept the tally the whole way through. Every number on this page comes from that one run. I did not do it twice and pick the better one — that would be cheating, and it would also spoil the point.

Here is the tally at four moments.

One real run of dice rolls, tallied at four moments A bar chart of the totals two to twelve. Thin dashed outlines show the share each total is predicted to get, rising to a peak at seven, and they never move. Solid bars show what one real run of three thousand six hundred rolls actually produced. After twelve rolls the solid bars are wildly uneven and seven has none at all. After sixty they are lumpy but the middle leads. After six hundred they form a dented hill. After three thousand six hundred they sit inside the dashed outlines all the way across. 23456789101112 after 12 rolls after 60 rolls after 600 rolls after 3,600 rolls dashed = predicted · solid = what the dice did

Step through one real run. The thin outline is the prediction from the thirty-six rolls, and it never moves. The bars are what the dice actually did.

  1. After 12 rolls. This is a mess. Seven has not come up at all, four and eight and ten and eleven are tied at the top, and the hill is nowhere. If this were all you had, you would say the prediction was wrong.
  2. After 60 rolls. Seven is out in front now, with 13 of them — more than the prediction expects. Nine is running second when it should be fourth. Lumpy, but the middle is clearly beating the ends.
  3. After 600 rolls. A hill, with dents. Six overshot and seven undershot, so the peak is in slightly the wrong place. Every bar is now within a few steps of its outline.
  4. After 3,600 rolls. Seven is back on top at 16.3% against a predicted 16.7%. Two is at 2.9% against 2.8%. The bars have crawled inside the outline and stayed there.

What settled, and what did not

Watch that again and notice which number is doing the settling. It is the share — the percentage — and not the count.

The count of sevens did not get closer to anything. It kept climbing: 0, then 13, then 87, then 586. It was always going to keep climbing, because more rolling makes more sevens.

What settled is 586 out of 3,600, which is 16.3%, against a prediction of 16.7%. That number stopped wandering. Early on it lurched about — 0%, then 21.7%, then 14.5% — and by 3,600 it was pinned within half a percentage point and staying there.

So the honest way to say what the counting predicts is not "you will get a seven every six rolls". It is "the share of sevens settles near one in six, and settles harder the longer you go".

Why more rolls help

There is a plain reason, and it needs no mathematics beyond dividing.

A single strange roll changes a small tally a lot and a big tally hardly at all. One extra seven out of 12 rolls moves the share by 8.3 percentage points. One extra seven out of 3,600 moves it by 0.03. The roll is the same size; the pile it lands on is not.

So the wobbles do not stop happening. They keep happening at exactly the same rate for ever. They just stop mattering, because each one is a smaller and smaller slice of the total.

One extra seven added to three tallies of different sizes Three horizontal bars, one for a tally of twelve rolls, one for sixty and one for six hundred. Each bar starts at the same length, the share a total of seven is predicted to get. A darker piece on the end of each bar shows how much that share moves when one more seven is added. The piece is long on the twelve-roll bar, short on the sixty-roll bar, and barely wider than a line on the six-hundred-roll bar. 12 rolls+8.33 points60 rolls+1.67 points600 rolls+0.17 points share of rolls that were a seven the dark piece is ONE extra seven, in every row
The same single extra seven, dropped onto three different piles. On twelve rolls it is a step you can see across the room. On six hundred it is barely thicker than the line drawn round the bar.

The number you actually need

There is no single roll count at which the dice become trustworthy. It depends on how sharp an answer you want.

Each extra decimal place of accuracy costs about a hundred times more rolling than the one before it. That is the real shape of the bargain, and it is why nobody settles these arguments by rolling. They settle them by counting the thirty-six.

After 3,600 rolls, a total of 7 has come up 586 times against a prediction of 600. What does another 3,600 rolls most likely do to that shortfall of 14?

The thing that has not been explained

Look at that last quiz answer again, because it is doing something sneaky.

Across the whole run, seven finished 14 short of its prediction. Two finished ahead of its prediction. Six finished well ahead. Those gaps did not vanish as the rolling went on — at 600 rolls six was ahead by 12, and at 3,600 it was still ahead.

And yet the shares all ended up in the right place.

Both of those are true at once, which sounds impossible. If seven is behind and stays behind, how does its share ever get where it is supposed to be? And if the dice are going to fix it, when exactly do they start?

There is a word for what everyone assumes is happening here. The word is "due", as in seven is due. It is wrong, and it is wrong in a way that has cost people a great deal of money.