Assume a school assigns each student a random pin number for their login account

Question

Assume a school assigns each student a random pin number for their login account one student at a time. Each pin number is generated by drawing 4 digits from the set (0,1,2,,,9) four times with replacement. We are interested in the specific pin number “2786” where the digits are in this order. The probability of this pin number being generated for a single student is p = P(2)P(7)P(8)P(6) = (1/10)^4. Assume this is a very large school.

a) What is the probability in the class of 28 students that exactly one person has this pin number? Do not compute the answer explicitly.

b) How many students in the school do we expect to have been assigned a this pin number when “2786” is assigned for the first time? Do not compute the answer explicitly.

c) What is the probability that the 3rd duplicate of this pin number is given out to the 12,000th student? Use the negative binomial distribution and solve for P(X = 12000). Do not compute the answer explicitly.

d) Lets say, so far there have been 7000 students in the whole school who have been assigned a pin number but none of them was assigned that one. What is the probability that this pin number is assigned for the first time to the 7,500th student. Do not compute the answer explicitly

Explanation / Answer

a) What is the probability in the class of 28 students that exactly one person has this pin number? Do not compute the answer explicitly.

Probability is = 1/10

b) How many students in the school do we expect to have been assigned a this pin number when “2786” is assigned for the first time? Do not compute the answer explicitly.

Probability = 1 / 10 * 2786

how many? ? 278.6

c)

Probability is = 1 / 10 * 1/10 * 1/10 = 0.001

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