MAH-CET 2026 — Mathematics PYQ

To find the relationship between the probability of the intersection of two events $P(A\cap B)$ and their individual probabilities, we use the fundamental Addition Rule of Probability.

Step 1: The Addition Rule

For any two arbitrary events $A$ and $B$ in a sample space, the probability of their union is given by:

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Step 2: Rearranging for the Intersection

We can rearrange this formula to solve for the probability of the intersection $P(A\cap B)$:

$$P(A\cap B)=P(A)+P(B)-P(A\cup B)$$

Step 3: Applying Axioms of Probability

According to the fundamental axioms of probability, the probability of any event (including the union of two events) can never exceed $1$. Therefore:

$$P(A\cup B)\le 1$$

Step 4: Deriving the Inequality

If we multiply the inequality $P(A\cup B)\le 1$ by $-1$, the direction of the inequality flips:

$$-P(A\cup B)\ge -1$$

Now, substitute this back into our rearranged equation from Step 2:

$$P(A\cap B)=P(A)+P(B)-P(A\cup B)\ge P(A)+P(B)-1$$

Thus, we arrive at the final inequality:

$$P(A\cap B)\ge P(A)+P(B)-1$$