Question

A spyware is trying to break into a system by guessing its password. It does not give up until it tries 1 million different passwords. What is the probability that it will guess the password and break in if by rules, the password must consist of

(a) 6 different lower-case letters
(b) 6 different letters, some may be upper-case, and it is case-sensitive
(c) any 6 letters, upper- or lower-case, and it is case-sensitive
(d) any 6 characters including letters and digits

Answer 1

The probability of a spyware program breaking into a system depends on the complexity of the password rules. By calculating the total number of possible passwords based on given rules and comparing it to the number of guessing attempts (1,000,000), one can determine the probability for each scenario.

Step-by-step explanation:

The probability of a spyware program guessing a password correctly can be calculated by determining the total number of possible unique passwords and then seeing how many attempts the spyware has in comparison.

1. 6 different lower-case letters: There are 26 possibilities for each character, and because the letters must be different, the total number of possibilities is 26 * 25 * 24 * 23 * 22 * 21. Since the spyware makes 1 million (1,000,000) attempts, the probability of guessing correctly is 1,000,000 / (26 * 25 * 24 * 23 * 22 * 21).
2. 6 different letters, case-sensitive: There are 52 possibilities for each character (26 lower-case + 26 upper-case), and since letters must be different, the total number of possibilities is 52 * 51 * 50 * 49 * 48 * 47. So the probability is 1,000,000 / (52 * 51 * 50 * 49 * 48 * 47).
3. Case-sensitive combination of letters: Since letters can be the same and are case-sensitive, there are 52 possibilities for each character, for a total of 52^6 possible combinations. The probability is 1,000,000 / 52^6.
4. Any 6 characters including letters and digits: There are 62 possibilities for each position (26 lower-case + 26 upper-case + 10 digits), giving us 62^6 possible combinations. The probability is 1,000,000 / 62^6.

In all cases, the probability of the spyware breaking in is the quotient of the number of attempts made (1,000,000) and the total number of possible passwords for each scenario.