Question

An ethical hacker, also referred to as a white hat hacker, is an information security expert who systematically attempts to penetrate a computer system, network, application or other computing resource on behalf of its owners. The purpose of ethical hacking is to evaluate the security of and identify vulnerabilities in systems, networks or system infrastructure. It includes finding and attempting to exploit any vulnerabilities to determine whether unauthorized access or other malicious activities are possible.

An ethical hacker is required to randomly guess the correct pin code that consists of the number 0 through 9 that must be entered in the correct order to access a company system.
1. What is the probability that the ethical hacker will guess the pin code correctly on the first try?
2. There are many variations of this guess. Assuming the primary variation allows the ethical hacker to guess correctly if the four-digit in the number are selected in any order as long as they are the same four digits as set by the IT security of the company.
For example, if the ethical hacker picks four digits making the number 2376, then the he will guest it right if 2376, 3726, 6327, 7632, and so forth, are entered. Consider the following four different versions of his presumptions.
(a) All four digits are unique (e.g. 1234)
(b) Exactly one of the digits appears twice (e.g. 2334, 8185)
(c) Two digits each appear twice (e.g. 1212, 8855)
(d) One digit appears three times (e.g. 2226, 8188)
Find the probability that the ethical hacker will successfully guess the accurate pin code in the first try for each of these four situations.
Show the necessary steps and explanation for the four presumptions stated above.

Answer 1

Answer: What is the probability that the ethical hacker will guess the pin code correctly on the first try?

Since the pin code consists of 10 possible digits (0 through 9), the probability of guessing the correct pin code on the first try is 1 in 10. This is because there is only one correct pin code out of the 10 possible options.

Therefore, the probability is 1/10 or 0.1 (or 10%).

Calculating the probability for each of the four situations:

(a) All four digits are unique (e.g., 1234):

In this case, there are 10 options for the first digit, 9 options for the second digit (since it can't be the same as the first), 8 options for the third digit, and 7 options for the fourth digit. The total number of possible combinations is given by:

10 × 9 × 8 × 7 = 5040

So, there are 5040 possible four-digit combinations when all digits are unique.

The probability of guessing the correct pin code on the first try in this situation is 1 in 5040, or 1/5040.

(b) Exactly one of the digits appears twice (e.g., 2334, 8185):

In this case, we have two scenarios to consider:

Scenario 1: The repeated digit is the first digit:

The first digit can be chosen in 10 ways, the second digit (repeated) can be chosen in 9 ways, and the remaining two distinct digits can be chosen in 8 and 7 ways, respectively. So, the total number of possible combinations is:

10 × 9 × 8 × 7 = 5040

Scenario 2: The repeated digit is not the first digit:

The first digit can be chosen in 9 ways (excluding the repeated digit), the repeated digit can be chosen in 10 ways, and the remaining two distinct digits can be chosen in 8 and 7 ways, respectively. So, the total number of possible combinations is:

9 × 10 × 8 × 7 = 5040

Combining both scenarios, we get a total of 2 × 5040 = 10080 possible combinations when exactly one of the digits appears twice.

The probability of guessing the correct pin code on the first try in this situation is 1 in 10080, or 1/10080.

(c) Two digits each appear twice (e.g., 1212, 8855):

In this case, there are two scenarios to consider:

Scenario 1: The two pairs of digits are different:

The first pair of digits can be chosen in 10 ways, the second pair of digits can be chosen in 9 ways, and the order of the pairs can be switched. So, the total number of possible combinations is:

10 × 9 × 2 = 180

Scenario 2: The two pairs of digits are the same:

The pair of digits can be chosen in 10 ways, and the order of the digits can be switched. So, the total number of possible combinations is:

10 × 1 = 10

Combining both scenarios, we get a total of 180 + 10 = 190 possible combinations when two digits each appear twice.

The probability of guessing the correct pin code on the first try in this situation is 1 in 190, or 1/190.

(d) One digit appears three times (e.g., 2226, 8188):

In this case, there are two scenarios to consider:

Scenario 1: The repeated digit is the first digit:

The first digit can be chosen in 10 ways, and the remaining two distinct digits can be chosen in 9 and 8 ways, respectively. So, the total number of possible combinations is:

10 × 9 × 8 = 720

Scenario 2: The repeated digit is not the first digit:

The first digit can be chosen in 9 ways (excluding the repeated digit), and the repeated digit can be chosen in 10 ways. The remaining distinct digit can be chosen in 8 ways. So, the total number of possible combinations is:

9 × 10 × 8 = 720

Combining both scenarios, we get a total of 2 × 720 = 1440 possible combinations when one digit appears three times.

The probability of guessing the correct pin code on the first try in this situation is 1 in 1440, or 1/1440.

To summarize, the probabilities for each of the four situations are:

(a) All four digits are unique: 1/5040

(b) Exactly one of the digits appears twice: 1/10080

(c) Two digits each appear twice: 1/190

(d) One digit appears three times: 1/1440