The motivation of my thesis came from a problem I heard on Radiolab, a podcast distributed through National Public Radio. In the podcast, the two hosts asked the question, “What is the probability of flipping seven consecutive tails when flipping a coin a hundred times?” They approximated the probability to being 1/6. From a mathematical point of view, this seems like too simple of an answer because there are 2^100 cases one must consider.
In my thesis, I first go about finding an exact probability to this initial question. Afterwards, I show how one can answer a generalized version of this question, where the number of flips and the number of consecutive events are variable. Additionally, I show how to find the probability of consecutive heads or tails occuring.
By answering these questions, I learned calculation techniques using matrix multiplication. These methods are shared in the paper. Lastly, I go into some of the underlying mathematics in this matrix multiplication and how it relates to their related recursive sequences.