Animated Solution for Mathematics - Probability: The probability that $A$ speaks truth is $4/5$, while the probability for $B$ is $3/4$. The probability that they contradict each other when asked to speak on a fact is

Welcome, future IITians! Today, we are not just solving a probability problem; we are dissecting the very nature of truth and falsehood. We are given the probabilities of two witnesses, $A$ and $B$, telling the truth:

We need to find the probability that they contradict each other.

Let us visualize this using a grid. Imagine a rectangle with a width of $5$ units and a height of $4$ units, where the total area is $20$ square units. This grid represents every possible outcome of their testimony.

The horizontal axis represents $A$, and the vertical axis represents $B$. Because $A$ speaks the truth $4$ out of $5$ times, we shade $4$ columns for truth and $1$ for a lie. Similarly, for $B$, we shade $3$ rows for truth and $1$ for a lie.

What does it mean to contradict? It means they are not in sync. If $A$ is honest, $B$ must be deceitful. If $A$ is deceitful, $B$ must be honest.

Mathematically, we are looking for two mutually exclusive events: 1. Case $1$: $A$ is true AND $B$ is false, denoted as $A\cap \bar{B}$. 2. Case $2$: $A$ is false AND $B$ is true, denoted as $\bar{A}\cap B$.

Because these events are independent, the probability of the intersection is the product of the individual probabilities.

For Case $1$, we calculate:

For Case $2$, we calculate:

Since these two cases are mutually exclusive, we sum them up to find the total probability of contradiction:

The final probability of the witnesses contradicting each other is $\frac{7}{20}$.

Remember, in JEE Advanced, the key is not just the calculation, but the visualization. Whether you use the grid method or the algebraic method, always ensure you understand the 'why' behind the 'what'.