Chapter 4.4, Problem 36E

Keep your password safe: A computer password consists of eight characters. Replications are allowed.

1. a. How many different passwords are possible if each character may be any lowercase letter or digit?
2. b. How many different passwords are possible if each character may be any lowercase letter?
3. c. How many different passwords are possible if each character may be any lowercase letter or digit, and at least one character must be a digit?
4. d. A computer is generating passwords. The computer generates eight characters at random, and each is equally likely to be any of the 26 letters or 10 digits. Replications are allowed. What is the probability that the password will contain all letters?
5. e. A computer system requires that passwords contain at least one digit. If eight characters are generated at random, what is the probability that they will form a valid password?

To determine

Find the possible number of passwords.

Answer to Problem 36E

There are $2.82\times {10}^{12}$ passwords are possible.

Explanation of Solution

Calculation:

The password consists 8 characters. Replications are possible. Each character are either lowercase letters or digits.

The Fundamental Principle of Counting:

If an action can be perform in m ways and another action can be perform in n ways then both the action can be done together in mn different ways.

The password consist 8 characters. Each character are either lowercase letters or digits.

Each digit can arrange in 10 different ways and letters can arrange in 26 different ways. That is each character can be arranged in 36$\left(=26+10\right)$ different ways. Substitute this value in the fundamental principle of counting.

$\begin{array}{c}N\left(\text{Number passwords}\right)=36\times 36\times \mathrm{...}\times 36\\ ={36}^8\\ =2.82\times {10}^{12}\end{array}$

Thus, there are $2.82\times {10}^{12}$  passwords are possible.

To determine

Find the possible number of passwords if each character may be any lowercase letters.

Answer to Problem 36E

There are $2.09\times {10}^{11}$ passwords are possible if each character may be any lowercase letters.

Explanation of Solution

Calculation:

The number plate contains lowercase letters. Each letters can arrange in 26 different ways. Substitute these values in the fundamental principle of counting.

$\begin{array}{c}N\left(\text{Passwords}\right)=26\times 26\times \mathrm{...}\times 26\\ ={26}^8\\ =2.09\times {10}^{11}\end{array}$

Thus, there are $2.09\times {10}^{11}$ passwords are possible if each character may be any lowercase letters.

To determine

Find the possible number of passwords if each character may be any lowercase letters or digits, and at least one character must be digit.

Answer to Problem 36E

There are $2.61\times {10}^{12}$ passwords are possible if each character may be any lowercase letters or digits, and at least one character must be digit.

Explanation of Solution

Calculation:

Number of passwords with each character may be any lowercase letters or digits, and at least one character must be digit can be obtained by the formula:

$\text{Required number of password}=\left[\begin{array}{l}\text{number of password with each character is either }\\ \text{a letters}\text{or a digits}-\text{number of password}\text{with}\text{each }\\ \text{character is letters}\end{array}\right]$

From part (a), it is clear that there are $2.82\times {10}^{12}$ passwords with each character are either lowercase letters or digits.

From part (b), it is clear that there are $2.09\times {10}^{11}$ passwords with each character may be any lowercase letters.

Substitute these values in the above equation.

Therefore, required number is,

$\begin{array}{c}N\left(\text{Passwords}\right)=2.82\times {10}^{12}-2.09\times {10}^{11}\\ =2.61\times {10}^{12}\end{array}$

Thus, there are $2.61\times {10}^{12}$ passwords are possible if each character may be any lowercase letters or digits, and at least one character must be digit.

To determine

Find the probability that the password will contain all letters.

Answer to Problem 36E

The probability that the password will contain all letters is 0.074.

Explanation of Solution

Calculation:

The probability of an event can be obtained as shown below:

$P\left(A\right)=\frac{\text{number of outcomes in }A}{\text{ number of outcomes in the sample space}}$

Event A denotes that the password will contain all letters.

From part (a), total number of password is $2.82\times {10}^{12}$.

From part (b), numbers of password will contain all letters is $2.09\times {10}^{11}$.

Substitute $2.09\times {10}^{11}$ for “number of outcomes in A” and $2.82\times {10}^{12}$ for “number of outcomes in the sample space”

The required probability is obtained as follows:

$\begin{array}{c}P\left(A\right)=\frac{2.09\times {10}^{11}}{2.82\times {10}^{12}}\\ =\frac{2.09}{28.2}\\ =0.074\end{array}$

Thus, the probability that the password will contain all letters is 0.074.

To determine

Find the probability that the password contain at least one digit form a valid password.

Answer to Problem 36E

The probability that the password contain at least one digit form a valid password is 0.9260.

Explanation of Solution

Calculation:

Event B denotes that the password contain at least one digit.

From part (a), total number of password is $2.82\times {10}^{12}$.

From part (c), numbers of password will contain at least one digit is $2.61\times {10}^{12}$.

Substitute $2.61\times {10}^{12}$ for “number of outcomes in B” and $2.82\times {10}^{12}$ for “number of outcomes in the sample space”

The required probability is obtained as follows:

$\begin{array}{c}P\left(A\right)=\frac{2.61\times {10}^{12}}{2.82\times {10}^{12}}\\ =\frac{2.61}{2.82}\\ =0.9255\\ =0.926\end{array}$

Thus, the probability that the password contain at least one digit form a valid password is 0.9260.