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?
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.