This problem seems to come up in various forms.

• What are the chances of a RPG character who can make multiple attacks of landing at least one blow?
• Using a dice-pool/success-counting mechanic, what are the chance of getting at least one success when rolling n dice?

What are the chances of rolling at least one 6 on n dice?

The probability of rolling one 6 on one die is 1:6.

The combined probability of a:b with x:y is...

by - (b - a)(y - x) : by

...or...

ay + bx - ax : by

Which you can visualize thusly...

So, the probability of at least one 6 on 2 dice is...

1(6) + 6(1) - 1(1) : 6(6)

11:36

The probability of at least one 6 on 3 dice is thus...

11(6) + 36(1) - 11(1) : 36(6)

91:216

The probability of at least one 6 on 4 dice...

91(6) + 216(1) - 91(1) : 216(6)

671:1,296