what is the probability of rolling a 4 at least once after 7 rolls?

Thanks Geno - that is by far the easiest way :)

I would call the probability of not thowing any 4 in seven rolls and the probabiliy of throwing at least one 4 in seven rolls a conjugate pair because on or the other has to happen (there must be a word for this) and they cannot both happen.  (They are mutually exclusive).

I have just googled this to see if my terminology is correct but all I got was a heap of 'gobble'd'g**k'.

Can a mathematician please discuss the correct terminology here.   

The probability of rolling at least one 4 plus the probability of rolling no 4s is 1.000.

It is easier to calculate the probability of rolling no 4s and subtracting that value from 1.000 to get the probability of rolling at least one 4 than it is to calculate the probability of rolling at leawst one 4 directly.

Probability of rolling no 4's:   --->   the probability of rolling anything but a 4 is 5/6. This probability holds for each roll, so if you roll a die 7 times, the probability of never getting a 4 is:  5/6 x 5/6 x 5/6 x 5/6 x 5/6 x 5/6 x 5/6. Now, subtract this answer from 1.000 and the result is the probability of rolling at least one 4.