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Why Is Seven the Commonest Roll?
Two dice do not give every total the same chance, because some totals have more routes.
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Step 1 of 5
One die is genuinely fair.
Six faces, six equal chances. Nothing about a single die favours any number, and that part everyone gets right.
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Step 2 of 5
Two dice give thirty-six pairs.
Six choices for the first, six for the second. Every one of those thirty-six pairs is exactly as likely as every other.
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Step 3 of 5
But totals collect different numbers of pairs.
Only one pair makes two — a one and a one. Six different pairs make seven. So seven turns up six times as often.
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Step 4 of 5
The shape it makes is a triangle.
Plot how often each total appears and you get a peak in the middle sloping away to both ends. Middles are common; edges are rare.
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Step 5 of 5
Adding more dice sharpens the peak.
With three dice the middle totals get commoner still and the extremes get rarer. Averaging anything pulls it toward the middle.
Board games put the best squares at distances of seven for exactly this reason.
The short version
Seven is commonest because six different pairs make it, while two and twelve have only one route each.
Try it yourself
Roll two dice thirty-six times and tally the totals. Seven should lead, and the ends should be nearly empty.
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