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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?