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For any problem, is the probability that at least 3 out of 10 people like doing something and the probability that at least 30 out of 100 people like doing something the same? I was wondering if as it grew larger, the probability would grow or shrink. Is it exponential?

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    $\begingroup$ Well, why not compute both and compare? You can use a binomial distribution to do it exactly, or you can use the normal approximation to the distribution to give a (very good) approximation. $\endgroup$
    – lulu
    May 12, 2018 at 17:46
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    $\begingroup$ What's the answer to your question for $1$ out of $2$ people compared with $2$ out of $4$? You should be able to answer that by listing all the cases. $\endgroup$ May 12, 2018 at 17:54
  • $\begingroup$ For the first case I would recommend to use the binomial distribution since the approximation is not very accurate. And additionally the calculation of $P(X\geq 3)=1-P(X\leq 2)$ is not very time consuming. But at the second case it is a good idea from lulu to use the normal approximation. $\endgroup$ May 12, 2018 at 18:05

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You are probably assuming that each person independently either does it or not with some probability $p$. Fluctuations get smaller as the sample size goes up, so if $p \gt 0.3$ you would expect more chance for $30$ out of $100$ and if $p \lt 0.3$ you would expect more chance for $3$ out of $10$. If $p$ is just slightly above $0.3$ you might get a reversal because of the granularity of cases with $10$ people.

If $p=0.3$ the probability will be close to $0.5$, a little higher because you accepted exactly $0.3$ of the people doing it. It will be higher for $3$ in $10$ because the chance of exactly $3$ of $10$ is higher than the chance of exactly $30$ of $100$ due to granularity.

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