Here is an experiment that illustrates the difference between a single bin and multiple bins. Run a computer simulation for flipping 1,000 fair coins. Flip each coin independently 10 times. Let’s focus on 3 coins as follows: c_{1} is the first coin flipped; c_{rand} is a coin you choose at random ; c_{min} is the coin that had the minimum frequency of heads (pick the earlier one in case of a tie). Let \nu_{1}, \nu_{rand}, \nu_{min} be the fraction of heads you obtain for the respective three coins.
a) What is \mu for the three coins selected?
b) Repeat this entire experiment a large number of times (e. g., 100,000 runs of the entire experiment) to get several instances of \nu_{1}, \nu_{rand} and \nu_{min} and plot the histograms of the distribution of \nu_{1}, \nu_{rand} and \nu_{min}. c) Notice that which coins end up being c_{rand} and c_{min} may differ from one run to another.
d) Using (b), plot estimates for \mathbb{P} \left [ \left | \nu - \mu \right | > \epsilon \right ] as a function of \epsilon, together with the Hoeffding bound 2e^{-2\epsilon^{2}N} (on the same graph).
e) Which coins obey the Hoeffding bound, and which ones do not? Explain why.
f) Relate part (d) to the multiple bins