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:

Solution 141E

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:

a).

Let us define events are:

A = the event that password consists only letters.

B = the event that password consists only integers.

Now,

The total number of characteristics = 26 + 26 + 10

= 62.

The possible number of 8 digit password = [Because digits are independent to each other]

The number of characteristics in letters = 26 + 26

= 52.

The possible number of 8 digit letters password =

The number of characteristics in Integers = 10

The possible number of 8 digit Integer password =

Conside,

…..(1)

Where,

Hence,

Similarly,

[Because A and B are mutually Exclusive events ]

Finally,

0

Hence,

Substitute these values in equation (1), then we get:

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