How it works...

We run several simulations, using different sample sizes. For each sample size, we take 1,000 samples and we get the mean, and 97.5% and 2.5% percentiles. We use them to construct a 95% confidence interval. It is evident that the greater the sample sizes are, the narrower these bands will be. We can see that when taking just 10 elements the bands are extremely wide, and they get narrower quite quickly when we jump to a larger sample size.

In the following table you can see why, when elections are tight (nearly 50% of voters choosing A), they are so difficult to predict. Since the election is won by whoever has the majority of the votes, and our confidence interval ranges from 0.45 to 0.55 (even when taking a sample size of 500 elements), we can't be certain of who the winner will be.

Take a look at the following results—95% intervals—left simulated ones; right Gaussian approximation: