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How can I exploit symmetry to intuit P(A wins) + P(B wins) = 1, without performing algebra? 

I'm not asking about algebra that I can execute. How can I intuit P(A wins) + P(B wins) = 1 most quickly, without algebra? Any 15 year old can calculate that for the 

$p = 1/2$ case, $\dfrac{i}{N} + {\color{limegreen}\dfrac{N - i}{N}} = 1$. 

$p \neq 1/2$ case, $\dfrac{1 - (q/p)^i}{1 - (q/p)^N} + {\color{limegreen}\dfrac{1 - (p/q)^{N - i}}{1 - (p/q)^N}} = 1$. 



![Image alt text](https://math.codidact.com/uploads/YWzKei1TbXde4CohGaMtybEc)

Blitzstein. *Introduction to Probability* (2019 2 ed). p 73.