Animated Solution for Mathematics - Probability: Comprehension Passage

Welcome, future engineer! Today, we are not just solving a probability problem; we are stepping into the shoes of a football analyst. Imagine you are standing on the sidelines, watching two teams, $T_1$ and $T_2$, battle it out over two intense games.

Our goal is to predict the outcome of their total points, $X$ and $Y$. Probability is often intimidating because it feels like guessing, but in the JEE Advanced arena, it is actually about systematic counting.

First, we must respect the individual game. We are given the probabilities for $T_1$: a win ($W$) with $P(W)=\frac{1}{2}$, a draw ($D$) with $P(D)=\frac{1}{6}$, and a loss ($L$) with $P(L)=\frac{1}{3}$.

Since the two games are independent, the probability of any sequence of two games is simply the product of their individual probabilities. Think of this as a $3\times 3$ grid of possibilities. There are $3\times 3=9$ total outcomes.

The first challenge is to find the probability that $T_1$ scores more points than $T_2$. We need to hunt for the scenarios where $X>Y$.

The most dominant scenario is $WW$ (Win-Win). Here, $T_1$ earns $3+3=6$ points, while $T_2$ earns $0+0=0$ points. The probability is:

Consider $WD$ (Win-Draw) and $DW$ (Draw-Win). In both cases, $T_1$ earns $3+1=4$ points, and $T_2$ earns $0+1=1$ point. Since $4>1$, these are valid.

The probability for $WD$ is $\frac{1}{2}\times \frac{1}{6}=\frac{1}{12}$, and for $DW$ it is $\frac{1}{6}\times \frac{1}{2}=\frac{1}{12}$. Adding these up, we get:

Now, the second challenge: when do the teams tie? We need $X=Y$.

One path is $WL$ (Win-Loss) or $LW$ (Loss-Win). In both, $T_1$ gets $3+0=3$ points and $T_2$ gets $0+3=3$ points. The probability for $WL$ is $\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}$, and for $LW$ it is $\frac{1}{3}\times \frac{1}{2}=\frac{1}{6}$.

What about $DD$ (Draw-Draw)? Here, $T_1$ gets $1+1=2$ points and $T_2$ gets $1+1=2$ points. This is a valid tie with probability:

Summing these, we get:

The math is not just about numbers; it is about accounting for every single possibility without bias. By breaking the problem into independent events and summing the mutually exclusive cases, we turned a complex scenario into a simple, logical flow.

Keep this systematic approach in your toolkit, and no probability problem will ever intimidate you again. Keep pushing, keep learning!