I have a bag and in it are two colors of marbles; red ($r$) and green ($g$). The population is very large $>10^{30}$. I would like to know with some level of confidence if there are more red marbles in the bag than green marbles by drawing a sample of $n$ marbles out. Obviously $P(g)=\dfrac{n_{green}}{n_{drawn}}$ and $P(r)=1-P(g)$.
I want to determine the confidence that there are more red marbles than green.
I think this is might be a null-hypothesis question in that I can test the hypothesis that $P(r)>P(g)$ but I am a bit out of practice in probability and statistics.
For example if I draw $20$ marbles and $17$ are red, then I'd be confident $P(r)>P(g)$. However, with $100$ draws giving $55$ red, I might be less confident---even though I've had more draws.
The null hypothesis (null = no difference) is that there is no significant difference in the proportion of red to green marbles. The alternate hypothesis is that there are more red than green.
The larger the sample, the more powerful the test and the less chance there is of making an error. For a given population parameter, a larger sample is always better. The likelihood of drawing $17$ out of $20$ and then $55$ out of $100$ for a given single population proportion is small but even so, the result of a one proportion Z test is..................... $P(g1) = .0009$ for $17$ out of $20$ and $P(g2) = .1587$ for $55$ out of $100$ maybe suggesting they are different populations.
At the 95% confidence level, you would reject the null hypothesis in the first instance and fail to reject it in the second.