# Chance of winning against improving winning rate

I. Player A and B is about to play a game G. We don't have any information about the players. I assume there is 50-50% chance who wins.

II. Let's say player A wins. How would it change his 50% for the next game? (We still don't have any other information eg. how did he play, did he have 'luck' etc.)

III. Player A and B has a history of playing game G. A leads 12-8 against B. What are the chances for A to win against B in their next game?

IV. Same situation as in III. but we know the distribution of A's games. (1 = win, 0 = lose) 1-1-1-1-0-1-1-1-0-1-1-1-0-1-0-1-0-0-0-0. (This should represent an improving performance of player B against A) Does this information change III. result?

This are more theoretical questions. I do not know if they are mathematically calculable. What are the keywords if I want to learn more about this kind of statistic / mathematics problems?

EDIT

I tried to formulate my questions as theoretical as possible, but I think it is easier to answer them if I define the game G as a sport, eg 1 vs 1 basketball. So the players have skill, they can train / practice to getting better. It is not like they would play coin flip against each other.

• This could be regarded as essentially a question about Bayesian statistics (you have some prior belief about their relative skill - some representation consistent with your stated ignorance - which you update as you get data). Jul 3, 2018 at 3:06

Since you have also tagged this question with the game-theory tag, it is worth noting that this field looks at optimal actions in games based on their internal structure. If you would like to analyse the game from this perspective, then you will need to formulate its internal structure and then look at the optimal behaviours using standard appeals to Nash equilibria, subgame perfect equilibria, or evolutionary equilibria. Since you are dealing with a sequence of games this would involve looking at the literature on repeated games.
• game-theory tag was maybe a mistake. The game in my examples is more like sport. (eg. 1vs 1 basketball) Jul 3, 2018 at 14:31