Probability with Catalan number with more open parenthesis

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Two candidates A and B raced each other in an election campaign. Candidate A accumulated a votes and candidate B accumulated b votes. Assuming a > b what is the probability that candidate A was leading all the way?

the answer is surprisingly $ \frac {a-b}{a+b} $

I had tried a few things such try to count all the vectors with 2b parenthesis and rearrange the $a-b$ rest parenthesis. tried to define a few recursive sequences. but nothing works.

thanks in advance