A computer system uses passwords that are exactly six characters and each character is one of the 26 letters (a–z) or 10 integers (0–9). Suppose that 10,000 users of the system have unique passwords. A hacker randomly selects (with replace- ment) 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?

The number of hits would follow a binomial distribution with and .

The probability of finding hits is approximately (or equivalently, approximately .)

The mean of the number of hits is approximately . The variance of the number of hits is approximately (not the same number as the mean.)

Step-by-step explanation:

There are possible passwords in this set. (Approximately two billion possible passwords.)

Each one of the randomly-selected passwords would have an approximately chance of matching one of the users' password.

Denote that probability as :

.

For any one of the randomly-selected passwords, let denote a hit and denote no hits. Using that notation, whether a selected password hits would follow a bernoulli distribution with as the likelihood of success.

Sum these 's and 's over the set of the randomly-selected passwords, and the result would represent the total number of hits.

Assume that these randomly-selected passwords are sampled independently with repetition. Whether each selected password hits would be independent from one another.

Hence, the total number of hits would follow a binomial distribution with trials (a billion trials) and as the chance of success on any given trial.

The probability of getting no hit would be:

.

(Since is between and , the value of would approach as the value of approaches infinity.)

The mean of this binomial distribution would be:.

The variance of this binomial distribution would be:

.

l.ln.lln.k.nkl

step-by-step explanation:

answer is x<