How likely is at least one six when you roll several dice?

Count the misses first

For a fair six-sided die, five of the six faces miss your target. With independent rolls, the chance that every die misses is (5/6) raised to the number of dice. Subtract that from 1 to find the chance of at least one six.

For two dice, no six has probability 25/36. The remaining 11/36, about 30.56%, contains a six on the first die, the second die or both. This is our worked application of the independent-trials model explained by OpenStax.

Source: OpenStax · The Binomial Distribution

Six dice still miss quite often

For six dice, the answer is 1 − (5/6)⁶, or about 66.51%. That leaves about 33.49% for no six. So roughly one third of such rolls miss entirely under this model.

The expected number of sixes is one, but results with zero, two or more sixes are all possible. The expectation averages across repetitions; it does not reserve one successful face in each handful of dice.

Exactly one is a different question

With two dice, exactly one six has probability 10/36, about 27.78%. Both dice showing six adds another 1/36 to the at-least-one result. Make this distinction before calculating a game rule.

For n fair s-sided dice and one target face, the general at-least-one formula is 1 − ((s − 1)/s)ⁿ. Four d12 dice therefore hit a particular face with probability about 29.39%. A rule such as 10 or higher on d12 has three target faces, so it needs a different success probability.

Use a roll to explore, not to prove the percentage

In Picknado's Dice Roller, choose the number of dice and enter their side count. Each displayed face is a separate draw; the total is their sum. A total of six is not the same event as one face showing six.

Try a short series of rolls and note the misses as well as the hits. Your observed percentage may differ sharply from the table over a small sample. Keep the same dice and target throughout the comparison; choosing a new rule after seeing the result changes the question.