Problem 44 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

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 (a) \(\frac{2}{9}\) (b) \(\frac{1}{9}\) (c) \(\frac{8}{9}\) (d) \(\frac{7}{9}\)

Step-by-Step Solution

Answer

The probability is \( \frac{1}{9} \).

Step 1: Define the Total Number of Choices

There are 3 houses available, and each of the 3 applicants can choose any of the 3 houses independently. Thus, the total number of ways the three applicants can choose the houses is given by multiplying the choices available for each applicant: \[ 3 \times 3 \times 3 = 27 \] So, there are 27 possible combinations of choices the three applicants can make.

Step 2: Calculate the Favorable Outcome

To have all three applicants apply for the same house, they must all choose the same one out of the 3 houses. There are exactly 3 possible scenarios for this: all choose house 1, all choose house 2, or all choose house 3.

Step 3: Determine the Probability

The probability of an event is given by the ratio of the favorable outcomes to the total possible outcomes. We have 3 favorable outcomes (all three choosing the same house) and a total of 27 possible combinations. Thus,\[ \text{Probability} = \frac{3}{27} = \frac{1}{9} \]

Step 4: Conclusion

The probability that all three applicants apply for the same house is \( \frac{1}{9} \). Therefore, the correct option is (b) \( \frac{1}{9} \).

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that looks at how objects can be arranged and combined. It helps us understand the number of different ways something can be chosen or organized.
For example, if each person can choose from 3 houses, we explore all the different picking scenarios through combinatorics. In the given exercise, we found that there are 27 possible ways the applicants could choose houses.
This is calculated by multiplying the choices each person has, i.e., 3 choices for one person, 3 for the second, and 3 for the third. Thus:

• The formula used is: Total Choices = Choices per person raised to the power of the number of people: \( 3^3 = 27 \).

This approach is essential when trying to calculate probabilities, as it lays the groundwork for understanding how likely certain events are.

Independent Events

Independent events are those whose outcomes do not affect each other. In probability, understanding the concept of independence is critical.
In this exercise, each person's choice of house is an independent event. This means that the house an applicant chooses does not influence the other participants' choices.
Understanding independence is crucial because:

• It tells us that we can calculate the total number of combinations by simply multiplying the options available to each person.
• It ensures that probabilities calculated for different events do not overlap and mix with others, allowing for simpler calculations.

Favorable Outcomes

A favorable outcome refers to scenarios that match the conditions of our problem or question. In our situation, we are interested in the applicants applying for the same house.
Out of all possible combinations, scenarios where all applicants choose the same house are what we refer to as favorable outcomes.
In the exercise:

• Regardless of the number of applicants, only three scenarios qualify as favorable: all choosing house 1, all choosing house 2, or all choosing house 3.
• These favorable outcomes directly impact how we subsequently calculate the probability of these events occurring.

Ratio of Outcomes

The concept of the ratio of outcomes is central in determining probabilities. Basically, this concept involves using the number of favorable outcomes divided by the total number of possible outcomes.
In our exercise, this ratio is crucial to finding the probability of all applicants choosing the same house. We counted three favorable outcomes, as previously discussed, out of 27 possible scenarios.
Thus, the probability is calculated as:

• Probability = Favorable Outcomes / Total Possible Outcomes = \( \frac{3}{27} = \frac{1}{9} \)

Understanding this ratio is vital in probabilistic problems because it provides a clear numerical expression of how likely a particular scenario is compared to all possible scenarios.