computer system uses passwords that contain exactly eight characters, and each character is one of the 26 lowercase letters (a–z) or 26 uppercase letters (A–Z) or 10 integers (0–9). Let ? denote the set of all possible password, and let A and B denote the events that consist of passwords with only letters or only integers, respectively. Suppose that all passwords in ? are equally likely. Determine the following robabilities:
Step1 of 4:
From the given problem we have:
The number of characteristics contain a password = 8,
Lower case letters = 26,
Upper case letters = 26, and
Integers = 10.
Here our goal is:
a). We need to find
b). We need to find
c). We need to find the probability of password contains exactly 2 integers given that it contains at least 1 integer.
Step2 of 4:
Let us define events are:
A = the event that password consists only letters.
B = the event that password consists only integers.
The total number of characteristics = 26 + 26 + 10
The possible number of 8 digit password = [Because digits are independent to each other]
The number of characteristics in letters = 26 + 26
The possible number of 8 digit letters password =
The number of characteristics in Integers = 10
The possible number of 8 digit Integer password =
[Because A and B are mutually Exclusive events ]
Substitute these values in equation (1), then we get: