Animated Solution for Mathematics - Probability: If $A$ and $B$ are two events such that $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{5}$ and $P(A\cup B)=\frac{1}{2}$, then $P(A|B^{\prime })+P(B|A^{\prime })$ is equal to

A Probability Journey

Probability is not just about rolling dice or flipping coins; it is the mathematical language we use to navigate the unknown. Today, we are going to dissect a classic JEE Advanced problem that tests your ability to bridge the gap between set theory and conditional logic.

We are given two events, $A$ and $B$, with $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{5}$, and their union $P(A\cup B)=\frac{1}{2}$. Our mission is to calculate the sum $P(A|B^{\prime })+P(B|A^{\prime })$.

Finding the Hidden Overlap

Before we can talk about conditional probabilities, we must understand the relationship between $A$ and $B$. Imagine a Venn diagram where we have two circles, $A$ and $B$, partially overlapping.

The Addition Theorem is our most powerful tool here. It states that the probability of the union is the sum of the individual probabilities minus their intersection:

By rearranging this, we can isolate the intersection, which represents the 'shared' reality of both events:

Substituting our known values, we get:

To solve this, we find a common denominator, which is $30$. Thus:

This tiny value, $\frac{1}{30}$, is the heart of our problem. Without it, the rest of the puzzle remains locked.

Decoding the Conditional

Now, let us tackle the first part of our objective: $P(A|B^{\prime })$. This notation asks: "What is the probability that $A$ occurs, given that $B$ has not occurred?"

By definition, this is:

The numerator, $P(A\cap B^{\prime })$, represents the region of $A$ that exists strictly outside of $B$. Geometrically, this is the total probability of $A$ minus the intersection we just calculated:

The denominator, $P(B^{\prime })$, is the complement of $B$, which is $1-P(B)=1-\frac{1}{5}=\frac{4}{5}$. Putting these together, we find:

Symmetry and Completion

Now, we turn to $P(B|A^{\prime })$. By the same logic of symmetry, this is the probability that $B$ occurs given that $A$ has not occurred:

The numerator is the region of $B$ outside $A$:

The denominator is the complement of $A$:

Thus, the second conditional probability is:

Finally, we sum our two results:

The elegance of this result is a testament to the consistency of probability theory. We started with three simple values and, through logical deduction, arrived at the final answer of $\frac{5}{8}$.