A computer system uses passwords that are exactly six...

A computer system uses passwords that are exactly six characters and each character is one of the 26 letters $(\mathrm{a}-\mathrm{z})$ or 10 integers $(0-9)$. Suppose that 10,000 users of the system have unique passwords. A hacker randomly selects (with replacement) one billion passwords from the potential set, and a match to a user's password is called a hit. (a) What is the distribution of the number of hits? (b) What is the probability of no hits? (c) What are the mean and variance of the number of hits?

A computer system uses passwords that are exactly six characters and each character is one of the 26 letters $(\mathrm{a}-\mathrm{z})$ or 10 integers $(0-9)$. Suppose that 10,000 users of the system have unique passwords. A hacker randomly selects (with replacement) one billion passwords from the potential set, and a match to a user's password is called a hit.
(a) What is the distribution of the number of hits?
(b) What is the probability of no hits?
(c) What are the mean and variance of the number of hits?

Key Concepts

Binomial Distribution

This is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has the same probability of success. In this context, each password guess is considered an independent trial that may either result in a hit (success) or not, with a constant probability of success over all trials.

Independent Trials

The concept of independent trials refers to the idea that the outcome of one trial does not affect the outcome of any other trial. Since the hacker selects passwords with replacement, each selection is independent, which is a necessary condition for applying the binomial distribution.

Probability of Success in a Trial

In problems like this, each trial (or guess) results in a success if a randomly chosen password matches one of the user passwords. The probability of success is calculated by dividing the number of favorable outcomes (the number of user passwords) by the total number of possible outcomes (the size of the entire password space). This probability remains constant across all trials.

Mean and Variance of a Binomial Distribution

For a binomially distributed random variable with n trials and success probability p, the mean (or expected value) is given by n*p, and the variance is given by n*p*(1-p). These formulas provide measures of the central tendency and spread of the number of hits expected over the trials.

Probability of Zero Successes

The probability of having no successes in a binomial experiment (i.e., zero hits) is computed by raising the probability of failure in a single trial (1-p) to the power of n (the total number of trials). This concept is used to determine the likelihood that none of the hacker's attempts match a user password.

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