The Ballot Theorem

Suppose in an election with two candidates, we are certain of the final tally. However, as the votes are counted sequentially, the lead may switch between the candidates during the process. This was certainly the case in the 2020 U.S. presidential election, where in several swing states, different types of votes (mail, early-voting, in-person) were counted at different stages. While the final tally does not change in a recount, the lead change may. One may ask: How likely will there be lead changes in a recount? Can the winning candidate always lead from start to finish? The Ballot Theorem provides a quantitative answer to this question. (Informal) Suppose candidates and each have […]

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Normal numbers

Imagine if we sample numbers from the unit interval and count the occurrences of each digit in every number’s infinite decimal representation. For instance, in the number , every non-zero digit appears exactly once, but appears infinitely often (trailing zeros). On the other hand, for the number , the digit occurrences (after the “dot”) is equal for every digit. As the number of samples increases, one may expect that the number of occurrences for each digit in to be close to one another. In fact, the Normal Number Theorem, first formulated by Borel, not only formalizes this intuition but goes a step further: for almost every number in , each digit appears […]

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