Consider a single bet with odds $o$ and thereby implied probability $1/o$.
Assume that the real probability $p$ is known.
Let $I$ be the stake, and $y$ the return from the bet. Then,
$\mathbb{E}(y) = I(po-1)$
Consider the case where you have $N$ such bets, and then define the win-ratio to be
$WR = \displaystyle\frac{\sum(return)}{\sum(bet)} = \displaystyle\frac{\sum_{i=1}^Ny_i}{\sum_{i=1}^NI_i}$.
My question is: would $WR$ be normally distributed by the Central Limit Theorem?