I'm currently reading the Wikipedia page proof on the Ballot Theorem, where one of the provided proofs was via induction.
They start off with two base cases, one where the first candidate receives all the votes and the other where both candidates receive zero votes. This part, I understand.
They then proceed to have two inductive hypotheses, which looks an awful lot like strong induction.
My question is, why is it that we need two base cases and two hypotheses here?
Since the induction step looks at the last vote, which could have gone to the first candidate or the second candidate, you need to look back to two previous results to complete it.
You could make it look more like weak induction on the total number of votes $n$ if your hypothesis was restated by saying "In an $n$-vote election where candidate A receives $p$ votes and candidate B receives the other $n-p$ votes with $p \gt \frac{n}{2}$, the probability that A will be strictly ahead of B throughout the count is $\frac{2p}{n} - 1$."
With this restated hypothesis, you would still need to look back to two different possibilities, but they would each be associated with the same value of $n$.