There are two games of chess, the first round and the second round are played against the same opponent of unknown skill.
$W_1$ is the event that round one is won by me.
$W_2$ is the event that round two is won by me.
$A$ is the skill of the opponent is known in general.
$A_1$ is the skill of the opponent is known in round one.
Given that I'm assuming the conditional probabilities are independent
$P(W_2|A)*P(W_1|A)=P(W_2,W_1|A)$ which means $P(W_2|W_1,A)=P(W_2|A)$
Why does the statement $P(W_2|W_1,A)$ seem incomplete? Shouldn't we be conditioning like this?
$P(W_2|W_1givenA_1,A)$
It's confusing in the sense that $P(W_2|W_1,A)$ seems incomplete due to the fact that I am not conditioning on $W_1$ given "knownledge of the player's skill in the first round" shouldn't that knowledge play a part in the shaping of the event of $W_1$ thus we should include $A_1$. I know that $P(event|A)$ is a function, but i'm more confused about how the knowledge of $A_1$ shapes the event $W_1$.
Perhaps, I'm confused about how events behave in the conditional part of the equation:
$P(event|"confused part")$ ?
I should mention that the skill of the opponent is invariant throughout the matches.