A survey is being conducted in a city with 1 million residents. It would be far

Question

A survey is being conducted in a city with 1 million residents. It would be far too expensive to survey all of the residents, so a random sample of size 1000 is chosen (in practice, there are many challenges with sampling, such as obtaining a complete list of everyone in the city, and dealing with people who refuse to participate). The survey is conducted by choosing people one at a time, with replacement and with equal probabilities. (a) Explain bow sampling with vs. without replacement here relates to the birthday problem. (b) Find the probability that at least one person will get chosen more than once.

Explanation / Answer

a) In this problem, people are sampled one by one, each person randomlyis any of the 10^6 residents in the city;

in the birthday problem, people show up ata party one by one, and each person randomly has any of 365 possible birthdays.when multiple pwople have same birthday

same person can be chosen more than once when replacement is anaglous

(b) the same structure as the birthday problem.fallows

p= (10^6(10^6-1)(10^6-2).....(10^6 -999)) / (10^6) 100

= ( 1- 1/10^6) (1-2/10^6) ....(1-999/10^6)

the probability of at least one match is 1-p0.393

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