What is the information entropy of the results of flipping a biased coin 1000 times, if the coin comes up tails with probability 5/6 and heads with probability 1/6? For this weighted coin, can you find an encoding for sequential pairs of coin flips (e.g. HH, HT, etc.) in terms of sequences of bits (possibly different numbers of bits for different possible pairs) that takes fewer than 2 bits on average for each pair of flips? How many bits on average does your encoding take, and how close is it to the information-theoretic limit?