Question 1183307: 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 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?
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! The number of possible passwords is 36^6=2176782336
With a billion password attempts the mean number of passwords one would get with the same probability each time (it is with replacement) is 1 billion/the above probability
That equals 0.4594 hits, n*p.
When n goes to infinity and p goes to 0 (but is not 0), the binomial distribution, which this is, can be approximated by a Poisson, where np=lambda. Dist is Bin~(1,000,000,000, 1 /36^6)
p(0)=e^(-0.4594)*0.4594^0/0!=0.6317
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The mean and the variance are equal in a Poisson distribution, and they are 0.4594 hits/hits^2 respectively.
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