Animated Solution for Mathematics - Probability: Let $A$ and $B$ be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE ?

Imagine you are standing in front of a vast, abstract canvas representing our sample space, $S$. Within this space, two events, $A$ and $B$, exist as distinct regions.

We are given their probabilities, $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$, and we are told they are independent.

This word, 'independent,' is not just a label; it is a profound mathematical relationship. It tells us that the occurrence of $B$ is completely irrelevant to the occurrence of $A$. Whether $B$ happens or not, $A$ remains indifferent.

Now, consider the event $B^{\prime }$, which is the complement of $B$, or 'not $B$.' If $B$ is independent of $A$, then logically, the non-occurrence of $B$ must also be independent of $A$.

This is a powerful tool in your JEE arsenal. If $A$ and $B$ are independent, then $A$ and $B^{\prime }$ are also independent.

This means that knowing $B$ did not happen gives us zero information about $A$.

Let us test this using the formal definition of conditional probability. We want to evaluate $P(A|B^{\prime })$, which is defined as:

The numerator, $P(A\cap B^{\prime })$, represents the probability that $A$ occurs AND $B$ does not occur. Because $A$ and $B^{\prime }$ are independent, we can express this intersection as a simple product:

Now, substitute this back into our conditional probability formula:

Look at the beauty of this expression! The term $P(B^{\prime })$ appears in both the numerator and the denominator.

As long as $P(B') eq 0$, which is true here since $P(B)=\frac{1}{6}$, we can cancel them out. The expression simplifies elegantly to:

We have arrived at a result that confirms our intuition. The conditional probability of $A$ given $B^{\prime }$ is simply $P(A)$.

Since we were given $P(A)=\frac{1}{3}$, we conclude that $P(A|B^{\prime })=\frac{1}{3}$.

The math is not just a set of rules; it is a language that describes the underlying structure of uncertainty. When you see 'independent' in a JEE problem, do not just reach for the formula; visualize the independence, trust the property of complements, and watch as the complex terms cancel out to reveal the simple truth.