Problem 40 - A Complete Resource Book in Mathematics for JEE Main

Question

Consider a set ' \(P\) ' containing \(n\) elements. A subset ' \(A^{\prime}\) of ' \(P^{\prime}\) is drawn and there after set ' \(P^{\prime}\) is reconstructed. Now one more subset ' \(B\) ' of ' \(P\) ' is drawn. Probability of drawing sets \(A\) and \(B\) so that \(A \cap B\) has exactly one element is (A) \((3 / 4)^{n} . n\) (B) \(n .(3 / 4)^{n-1}\) (C) \(n \cdot(3 / 4)^{n}\) (D) none of these

Step-by-Step Solution

Answer

The answer is (B) \( n \cdot (3/4)^{n-1} \).

Step 1: Understanding the Problem

We have a set \( P \) with \( n \) elements, and we draw two subsets \( A \) and \( B \) such that their intersection \( A \cap B \) contains exactly one element. We are to find the probability of this event and match it to the given options.

Step 2: Calculating Possible Ways to Pick Elements

For each element of \( P \), there are 4 possibilities: it can be in neither \( A \) nor \( B \), only in \( A \), only in \( B \), or in both. Hence, there are \( 4^n \) ways to form subsets \( A \) and \( B \) together.

Step 3: Calculating Successful Outcomes

We need one element, say \( x \), to be in both \( A \) and \( B \). Choose \( x \) from \( n \) elements, \( n \) choices. For the other \( n-1 \) elements, either belong to \( A \) or \( B \) but not both or neither. Thus, 3 choices per remaining element, giving \( 3^{n-1} \) ways.

Step 4: Computing the Probability

The probability of intersection \( A \cap B \) having exactly one element is calculated as the number of successful outcomes over total possible outcomes: \[ \frac{n \cdot 3^{n-1}}{4^n} = n \cdot \left( \frac{3}{4} \right)^{n-1} \]

Step 5: Matching with Provided Options

The calculated probability \( n \cdot \left( \frac{3}{4} \right)^{n-1} \) matches option (B). Therefore, the correct choice is option (B).

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It's a fundamental concept in statistics and mathematics that allows us to quantify uncertainty. In this problem, we are interested in the probability that the intersection of two subsets (drawn from a larger set) contains exactly one element.

To calculate probability, we use the formula:

• Probability = Number of Successful Outcomes / Total Number of Possible Outcomes

Here, the successful outcome is when the intersection of two subsets has exactly one element, and the total possible outcomes are all the ways we can form these subsets.

In our context, the total number of ways to choose subsets is determined by the possible inclusion of each element in either subset or neither, leading to a total of:

• Total Possible Outcomes = 4^n

Subset

A subset is a set that contains some or none of the elements of another set, which we refer to as its superset. If we have a set labeled as "P" with "n" elements, any collection of elements from "P" can be considered a subset.

In this particular exercise, we're selecting subsets from a set "P". Subset "A" is drawn and later reconstituted into "P", after which we draw a second subset, "B".

Key points to remember about subsets:

• A subset can be empty (contain no elements) or be equal to the original set.
• Each element of the original set can either be included in the subset or not; thus, a set with "n" elements has 2^n possible subsets.

Set Intersection

The intersection of two sets is a new set containing all the elements that are present in both original sets. Mathematically, the symbol for intersection is \( \cap \). In our problem, we are interested in the intersection of subsets "A" and "B".

Our goal is for the intersection \( A \cap B \) to have exactly one element. This requirement means that there needs to be precisely one item that is common to both subsets while all other elements are not shared.

Understanding set intersection allows us to:

• Identify common elements between two groups.
• Utilize set theory to solve complex problems involving multiple sets, such as determining overlaps.

Mathematics Problem Solving

Mathematics problem-solving involves identifying, breaking down, and methodically tackling a problem to find a solution. We use logical reasoning and mathematical theories.

This exercise demonstrated problem-solving through steps:

• Understanding the Problem: Recognizing the aim is to find the probability and the constraints (e.g., intersection with one element).
• Calculating Possible Ways: Estimating all ways subsets can be formed, using combinations of set elements.
• Solving the Equation: Developing the expression for probability: \[ \frac{n \cdot 3^{n-1}}{4^n} = n \cdot \left( \frac{3}{4} \right)^{n-1} \]
• Matching the Solution to Options: Comparing the solution to given options to find the correct one.

Effective problem solving often involves double-checking each step to ensure accuracy and logic.