A computer system uses passwords that contain exactly seven characters, and each

Question

A computer system uses passwords that contain exactly seven 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 Ohm 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 number of passwords in each of the following events. (a) Ohm (b) A (c) A' intersection B' (d) Passwords that contain at least 1 lowercase letter Determine the probability for each of the following: (e) Password contains at least 1 lowercase letter given that it contains only letters (f) Password contains only odd numbers given that it contains all numbers

Explanation / Answer

Here' the solution. Please drop a comment in case you want more help on the solution.

There are , in this password:

1. 1 to 26 lowercase letters

2. A-Z upper case 26 letters

3. 0-9 10 integers

So, 1 blank can have 26+26+10 = 62 variantions

If 1 blank can be filled with only letter ,then 26+26 = 52 variations possible, You can't fill the blank with any of the 10 numbers in this case.

a. Sigma = ?

This is basically the number of combinations possible. So, out of the given each blank can take (26+26+10) = 62 unique value, total no. of possobilities = 62*62...*62.. 7 times. = 62^7

b. A consists of passwords with only letters. So , each blank can take 52 ( 26upper+26lower) types of letter.

= 52...*52 ..7 times

= 52^7

c) A' intersect B' = This is the area which has no integers and no letters. beyond number and integers nothing can be filled, and hence, no event exist which satisfy this condition.

Hence, no. of possibilities = 0

d) Password that contain atleast 1 lower case = All possobilites - Password that contain no lower case

= 62^7 - 36^7 ( 36 = 26 upper case letters + 10 numbers )

e. P( password contains at least 1 lowercase letter given that it contains only letters) = Possiblites of atleast 1 lowercase letter / possibilites that password has only letters

= ( 52^7 - 26^7)/52^7 [ please note that only 52 = 26 lower + 26 upper case are taken into consideration here]

= 1- (1/2)^7

= .992

f. P( pwd has only odd numbers given that it contains only numbers) = ?

1, 3, 5, 7, 9 are odd

= So , 5^7/10^7 ( out of 10 numbers available , we have 5 odd numbers)

= (1/2)^7

= .0078

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