Since I decided to call this blog martingalemeasure it seems only fitting that the first post should be about probability; martingales in particular. In my favorite introductory book on measure theoretical probability, “Probablity with Martingales” by David Williams, we find an exercise in Chapter 10, which I paraphrase here:
Suppose a monkey is typing randomly at a typewriter whose only keys are the capital letters $latex A$ through $latex Z$ of the english alphabet. What is the expected (average) time it will take for the monkey to type the word $latex ABRACADABRA$?
This is not an easy problem. In fact it’s not entirely obvious that the average time is even finte! Williams expects the reader to solve it using the beautiful theory of martingales and in particular Doob’s optional-stopping theorem. We will calculate the result below, but stop short of a proof. (There are many proofs of this result…
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