Animated Solution for Mathematics - Probability: For two given events $A$ and $B,P(A\cap B)$ is

When you look at a problem involving two events $A$ and $B$, stop seeing them as abstract variables. Imagine them as two distinct territories on a map, where the sample space $S$ is the entire world.

The intersection $P(A\cap B)$ represents the shared territory where both events coexist. This is the fundamental region of interest in our analysis.

The bedrock of probability is the Addition Theorem. It defines the relationship between the union and the intersection of two events:

The subtraction of the intersection is necessary because simply adding the probabilities of $A$ and $B$ counts the shared region twice. To obtain the true total probability of the combined region, we must subtract the overlap exactly once.

By rearranging the Addition Theorem, we isolate our target variable, the intersection:

This equation serves as our master key. It demonstrates that the probability of the intersection is entirely dependent on the probability of the union.

In probability theory, every event is governed by the laws of the floor ($0$) and the ceiling ($1$). Since $P(A\cup B)\le 1$, we can derive the lower bound for the intersection:

This profound result indicates that if $A$ and $B$ are sufficiently large, they are mathematically forced to overlap. The intersection cannot be arbitrarily small.

Conversely, since $P(A\cup B)\ge 0$, we can derive the upper bound for the intersection:

This confirms that the overlap cannot exceed the sum of the individual probabilities.

We have successfully derived the constraints on the intersection using the fundamental axioms of probability. The probability of the intersection is bounded as follows:

JEE Advanced mathematics is not about memorizing formulas; it is about understanding the logical constraints of the universe. By mastering these bounds, you are building a robust mental model of reality.