I've come up with a problem that seems so simple but I could not find where the problem is.
Assume we have three elements $A$, $B$, and $C$. Each element is a random number from $(0,1)$. What is the probability that $A$ is the maximum element in this array?
I think it can be expressed as $P( \{A>B\} \cap \{A>C\} )$, where $\{A>B\}$ and $\{A>C\}$ are independent events. Then seems the probability is 1/4. But surely we know the probability should be 1/3.
Where's the problem and what is the correct approach to calculate this probability?