# Probability Of Winning A Game

A series of $$2014$$ games is being played between two players $$A$$ and $$B$$. There are no ties-- exactly one of the players wins in a game and the other loses. The first game is won by $$A$$, and $$B$$ wins the second game. In the consequent games, the probability of $$A$$ winning is equal to $$\dfrac{P}{P+Q}$$, where $$P$$ and $$Q$$ denote the number of games won by $$A$$ and $$B$$ respectively so far (for example, the probability of $$A$$ winning the third game is $$\dfrac{1}{2},$$ if $$A$$ wins the third game, the probability of $$A$$ winning the fourth game is $$\dfrac{2}{3},$$ etc). The probability that the series is tied, i.e. $$A$$ and $$B$$ both win exactly $$\dfrac{2014}{2}$$ games., can be expressed as $$\dfrac{a}{b},$$ where $$a$$ and $$b$$ are coprime positive integers. Find $$a+b+1$$.

Details and assumptions

• This problem is not original.
×