Let A and B be independent events. What is $P(A | (A \cup B))$? I have two alternative interpretations and in particular I would like to know why one is right and the other wrong (or why both are wrong).
This could be interpreted as $A \cup B$ has happened (i.e. if circles A and B are drawn within a box representing the probability space, filling in both A and B), and so A must have happened as the region A or B has happened, so $P(A | (A \cup B)) = 1$. I feel like this is wrong but am not really sure why.
Alternatively, this could be interpreted that we now restrict the probability space to $A \cup B$, but it has not crystalised into an event yet, in which case $P(A | (A \cup B)) = \frac{P(A)}{P(A \cup B}$.
Additionally, can anyone recomend any reading in which Bayesian rules are explored with set notation rather than the standard form and examples found in probability books? e.g. I am struggling how to interpret things such as $P(A | B \cap (A \cup B))$ when A and B are / are not independent.
Many thanks.