On a fair die every face is equally likely, so one d6 gives each number a 1/6 chance, or 16.67%. Rolls don’t remember each other, so to get the odds of two things both happening you multiply. Sums of several dice are not flat: middle totals have more ways to happen.
Every face, same chance
That’s the whole starting point. A fair d6 has six faces and no reason to prefer any of them, so each comes up 16.67% of the time. A d20 gives each face 5%. Bigger die, smaller slice, same idea.
“Fair” is doing work in that sentence. A die with a weight in it, or with pips drilled deep enough to shift the balance, isn’t fair. That’s what loaded dice are. Casino dice get machined to tight tolerances and filled with paint of the same density as the body for exactly this reason.
Dice have no memory
Roll four 6s in a row and the fifth roll is still 16.67% to be a 6. The die doesn’t know what happened. Believing otherwise is the gambler’s fallacy, and it costs people money at craps tables constantly.
Because rolls are independent, you multiply to get the chance of a run. Two 6s in a row is 1/6 times 1/6, which is 1/36 of 36 equally likely outcomes, or 2.78%.
The flip side is the “at least one” question, and multiplying handles that too, just backwards. Work out the chance of the thing never happening, then subtract from 100%. Rolling at least one 6 on two dice is easier to compute as “not zero 6s” than by counting cases.
Why sums aren’t flat
One die is uniform. Two dice are not, and the reason is counting.
Two d6 land in 36 equally likely ways. Only one of those makes a total of 2, and only one makes 12, so each sits at 2.78%. But 7 can be built six different ways, giving it 6/36 and a 16.67% chance. That gap is why 7 dominates craps.
Add a third die and the pile tightens further. Three d6 average 10.5, and a total of 10 shows up 12.5% of the time against 0.46% for a total of 3. The more dice you add, the more results crowd the center and the thinner the tails get.
Expected value
Expected value is the long-run average, and it’s the number to reach for when comparing options. One d6 has an expected value of 3.5. Two d6 come to 7, three to 10.5, a d20 to 10.5.
It’s an average, not a prediction. You will never roll 3.5 on a d6. Over a few hundred rolls, though, your running average will sit close to it, and that’s enough to decide whether a Farkle push is worth it or which damage die to pick.
Spread matters alongside the average. Three d6 have a standard deviation of 2.96, meaning most totals land within a few points of 10.5.
Below you’ll find the full 2d6 chart laid out total by total, a page on the 7 in particular, and a calculator for any dice combination you want to check. If you’d rather see the numbers happen than read them, roll some dice and watch how fast the middle fills up.
FAQ
What are the odds of rolling a 7 with two dice?
16.67%, or 6/36 of the 36 possible combinations. Six pairs make 7: 1-6, 2-5, 3-4, and each of those in reverse order. No other total has that many ways to happen, which is why 7 is the most common result.
Are dice rolls independent?
Yes. Each roll starts fresh, and a die has no way to store what it did last time. A streak of sixes doesn’t make the next six less likely, and a long drought doesn’t make one overdue. The gambler’s fallacy is believing otherwise.
Why is 7 more likely than 2?
Counting. Only one combination gives a total of 2, both dice showing 1, so it lands at 2.78%. Seven can be made six different ways, so it lands at 16.67%. Same dice, same fairness, different number of routes to the total.
What does expected value mean for dice?
The average result over many rolls. For one d6 it’s 3.5; for 2d6 it’s 7. You can’t roll the expected value on a single die, but it tells you what a long session will average out to, which is what matters for deciding between two options.
How do I calculate the chance of rolling two sixes?
Multiply the two chances together. Each die shows a 6 with probability 1/6, so both showing 6 is 1/36, or 2.78%. The same multiplication works for any two independent results on separate dice.