I have a dataset made of couples $(n_i,v_i)$ where $n_i$ denotes the number of times a game has been played, and $v_i$ the number of victories at the $i-$th day.
What is the best way to evaluate the probability $P$ of winning the game? (We can assume that winning the game does not depend on time).
My first thought is to evaluate $P$ as the total number of victories over the number of games, i.e. $P = \frac{\sum_{i=1}^t v_i}{\sum_{i=1}^t n_i} $
I then had the doubt though that I could evaluate $p_i = v_i / n_i$ and define $P$ as the mean of the individual probabilities: $ P = \frac{1}{t} \sum_{i=1}^t p_i $.
Somehow I feel this second approach is wrong, but can't entirely understand why.
How would you evaluate $P$ and why? Can you give me some links explaining how to evaluate reliable statistics?