Exercise 1.26

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.
\begin{enumerate}[label=(\alph*)]
\item Explain how sampling with vs. without replacement here relates to the birthday
problem.
\item Find the probability that at least one person will get chosen more than once.
\end{enumerate}

fifthist x · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item  When sampling with replacement, the probability of any sample of size 
$1000$ is $$\frac{1}{K^{1000}}$$ where $K$ is the size of the population. 
However, if sampling is done without replacement, then the probability is 
$$\frac{1}{K(K-1) \dots (K-1000+1)}$$ which is different from the result in 
the birthday problem.

\item $$P(A) = 1 - P(A^{c}) = 1 - \frac{K(K-1) \dots (K-1000+1)}{K^{1000}}$$ 
where $K = 1000000$.
\end{enumerate}

Exercise 1.26

Answers

Comments

Exercise 1.26

Answers

Comments

Add answer