Animated Solution for Mathematics - Probability: Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is

Mastering Probability

Welcome, future engineer. Today, we are not just solving a probability problem; we are peeling back the curtain on how independent events weave together to form a sample space. This problem is a foundational pillar for understanding combinatorics in JEE Advanced.

The Power of Independence

Imagine you are standing in a locality with three distinct houses: $H_1$, $H_2$, and $H_3$. Three people, $P_1$, $P_2$, and $P_3$, are about to make a decision. The crucial word here is independently.

This means $P_1$ does not care what $P_2$ does, and $P_3$ is acting entirely on their own whim. When we analyze the choices for $P_1$, we see they have $3$ options. Similarly, $P_2$ has $3$ options, and $P_3$ has $3$ options.

In the world of combinatorics, when events are independent, we do not add these choices; we multiply them. This is the Fundamental Principle of Counting. The total number of ways these three people can distribute their applications is:

Substituting our values, we get:

This $27$ represents our entire sample space. It is the universe of all possible outcomes.

Defining the Favorable Universe

Now, we shift our focus to the specific condition: "All three apply for the same house." We need to count the scenarios that satisfy this constraint. Let us visualize the favorable outcomes:

1. Everyone chooses $H_1$: This is the tuple $(H_1,H_1,H_1)$. There is exactly $1$ way for this to happen. 2. Everyone chooses $H_2$: This is the tuple $(H_2,H_2,H_2)$. Again, exactly $1$ way. 3. Everyone chooses $H_3$: This is the tuple $(H_3,H_3,H_3)$. Once more, exactly $1$ way.

Since these scenarios are mutually exclusive—meaning they cannot happen simultaneously—we sum them up to find our total favorable outcomes:

The Elegant Conclusion

We have arrived at the final step. Probability is defined as the ratio of favorable outcomes to the total possible outcomes. It is the measure of how often our desired event occurs within the vast sea of possibilities.

Plugging in our numbers:

When we simplify this fraction by dividing both the numerator and the denominator by $3$, we arrive at the final, elegant result:

In JEE Advanced, you will encounter problems where the number of people and houses increases to $n$ and $m$. If you understand that the total outcomes are $m^n$ and the favorable outcomes for "all choosing the same" are $m$, you can generalize this to:

Never rush the setup. Visualize the agents, define the independence, and the math will flow naturally. You have mastered this concept today. Keep that momentum going!