Can Two or more Events be Exhaustive and Independent?
Two Event Problem Formulation
Let $S$ be the sample space.
Let $A,B \subset S$ be two strict events with probability $p,q$ respectively.
The conditions are
- $P(A \cap B) = P(A)P(B) = pq$ [Independent]
- $P(A \cup B) = P(S) = 1$ [Exhaustive]
Do there exist such $A,B$? If yes, how do they look like?
Hints
Hint 1
Using both the conditions and the identity $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ to an equation in $p,q$.
Hint 2
Solve for $p,q$ to get that $p=1$ or $q = 1$. But, $A,B$ are strict events. Hence, not possible.
Food For Thoughts
- What if, there are $n$ events, and can there be such $n$ exhaustive and independent events?