We want to estimate the probability p of a coin by sampling. We took n samples where each sample is X_i. Together, assume each X_i are i.i.d., we have:
Since n\delta \in \Theta(n), it makes sense to use Pretty Chernoff Bound for i.i.d. Binomial.
Also notice that \delta grows as \frac{1}{\sqrt{n}}.
So we conclude [\frac{X}{n} - 0.043, \frac{X}{n} + 0.043] forms a 95\% confidence interval on the true p.
Of course there are many issues that come up in statistical parameter estimation. For example, it is not obvious how to get "independent", equally weighted samples.
Balls and Bins
We randomly distribute n balls in n bins. Assuming n is sufficiently large, we want to show with high probability, no bin will have more than \frac{3\ln n}{\ln \ln n} - 1 \in O(\frac{\ln(n)}{\ln \ln n}) balls.
Sufficiently Large: n \rightarrow \infty
With High Probability: Pr \geq 1 - \frac{1}{n} with high n
Note that we chose \frac{3\ln n}{\ln \ln n} - 1 to simplify calculation.
The reason we chose k = \frac{3\ln n}{\ln \ln n} - 1 because it is slower than \ln(n)