Let be the event that there is a tie. Let be the event that the first vote counted is in Trump's favor, and define analogously. By the Total Probability Formula, To simplify this, we remark that by Bayes' Theorem, so the expression above becomes I now claim that . To prove this, let be the set of strings of votes for which is first and a tie exists; define similarly. Consider a string . Note that since begins with an , the vote immediately preceding the last tie in the string must be a . (Oops I said first tie in the HW writing session D:) Thus, consider the string which consists of the sequence of votes applied in reverse. Now the string starts with a , so . Similarly, we can take a string and reverse it to get a string , so we have a bijection and the desired probability is .

Now remark that since (if there were no tie the number of votes for would always have to be greater than the number of votes for , which is a contradiction). Furthermore, since all vote sequences are evenly distributed, . Therefore