Acomputer system requires that passwords contain at least one digit. if eight characters are generated at random, and each is equally likely to be any of the 26 letters or 10 digits, what is the probability that a valid password will be generated?

Figure this out as though no digits are present. You can choose 8 of the 26 letters and that's it.
26^8 [you can allow repeating letters]  =  2.088 * 10^11
The total number of ways that can be chosen with letters and digits intermixed = 36^8

The probability of getting no numbers is 26^8 / 36^8 or (26/36)^8 = 0.07402
So the probability that there were be at least one digit is
1 - 0.07402 = 0.92598