A computer system uses passwords that contain exactly 7 characters, and each character is 1 of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let Ω denote the set of all possible passwords, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Determine the probability that a password contains all lowercase letters given that it contains only letters. Report the answer to 3 decimal places.

To calculate the probability the selected password is made out only of lower-case letters, if it's only letters, we have first to find out how many passwords could be formed with only letters and with only lower-case letters.

For lowercase letters, we can make this many passwords, since for each of the 7 characters, we can pick among 26 lowercase letters:

NLL = 26 * 26 * 26 * 26 * 26 * 26 * 26

In the same fashion, for the number of passwords consisting only of letters, we can pick among 52 letters for each each character (26 lower-case, 26 upper-case):

NOL = 52 * 52 * 52 * 52 * 52 * 52 * 52

We can rewrite NOL differently to ease our calculations:

NOL = (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26) * (2 * 26)

or

NOL = 26 * 26 * 26 * 26 * 26 * 26 * 26 * 2 * 2 * 2 * 2 * 2 * 2 * 2

Now we have to find out the probability a password containing only letters (NOL) is a password containing only lowercase letters (NLL). So, we divide NLL by NOL:

[tex]\frac{NLL}{NOL} = \frac{26 * 26 * 26 * 26 * 26 * 26 * 26}{26 * 26 * 26 * 26 * 26 * 26 * 26 * 2 * 2 * 2 * 2 * 2 * 2 * 2} = \frac{1}{2 * 2 * 2 * 2 * 2 * 2 * 2} = \frac{1}{2^{7} }[/tex]

The probability is thus 1/2^7 or 1/128 or 0,0078125

Which we are asked to round to 3 decimals... so 0,008 or 0,8%