Let's say I have a two-answer poll, such as:
In which direction do you stir a cup of coffee?
A: Clockwise
B: Counter-clockwise
and the results are 60%:40% A:B.
If only 5 people were polled, the chances that the "real" distribution is 50%:50% are quite high.
However, if one million people were polled, you could be reasonably sure that the distribution is not, in fact, 50%:50%, and there is a significant bias toward clockwise stirrers.
What is the correct way to quantify uncertainty/error/confidence estimation in this situation? How do we quantify how confident to be in a distribution elucidated from a poll question such as this?