Problem 55 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Suppose \(n\) people are asked a question successively in a random order and exactly 3 of the \(n\) people know that answer. If \(n>6\), the probability that the first four of those asked do not know the answer is (A) \(\frac{{ }^{n-4} C_{4}}{{ }^{n} C_{4}}\) (B) \(\frac{{ }^{n-3} C_{4}}{{ }^{n} C_{4}}\) (C) \(\frac{1}{{ }^{n} C_{4}}\) (D) none of these

Step-by-Step Solution

Answer

The probability is \( \frac{\binom{n-3}{4}}{\binom{n}{4}} \), which is option (B).

Step 1: Identify the Scenario

We have a total of \( n \) people, and among them, exactly 3 know the answer to a question. We want the probability that the first four asked do not know the answer. This means the first four must be among the \( n - 3 \) people who do not know the answer.

Step 2: Calculate Ways to Choose the First Four

Since the first four people do not know the answer, we select 4 people from the \( n - 3 \) who do not know the answer. The number of ways to do this is given by the combination \( \binom{n-3}{4} \).

Step 3: Calculate Total Ways to Choose Any Four

The total number of ways to choose any four people from the potential \( n \) people is given by \( \binom{n}{4} \). This represents all possible combinations of choosing four people.

Step 4: Calculate the Probability

The probability that the first four asked are among the \( n - 3 \) who do not know the answer is the ratio of favorable outcomes to the total outcomes. Therefore, the probability is \( \frac{\binom{n-3}{4}}{\binom{n}{4}} \).

Step 5: Compare with Given Options

Comparing our result \( \frac{\binom{n-3}{4}}{\binom{n}{4}} \) with the given options, we find that it matches option (B).

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics that focuses on counting, arranging, and analyzing sets and their elements. It plays a crucial role in probability theory by providing methods to count possible outcomes.
In our problem, combinatorics helps us determine the number of ways to select people from a group. This is done without concern for order, which is a common method known as combinations. By using combinations, we can efficiently calculate the different ways events can occur.

• Imagine you have a group of people and need to choose a subset. Combinatorics provides a systematic way to count all possible combinations.
• For any given number of people, say 10, and you need to choose 4, combinatorics allows you to find how many such groups of 4 can be formed.

This systematic approach is key in solving probability-related problems where the order does not matter.

Binomial Coefficients

Binomial coefficients are numerical values that describe the number of ways to choose a subset of a specific size from a larger set. They are represented by the notation \( \binom{n}{r} \), which means choosing \( r \) items from a set of \( n \) items.
In our exercise, binomial coefficients are used to determine two important figures:

• The number of ways to select 4 people from those who do not know the answer, represented as \( \binom{n-3}{4} \).
• The total number of ways to select any 4 people from the total group, represented as \( \binom{n}{4} \).

Binomial coefficients are computed using the formula:
\[ \binom{n}{r} = \frac{n!}{r!(n-r)!} \]
where \( n! \) is the factorial of \( n \), meaning the product of all positive integers up to \( n \). This calculation provides the total count of subsets possible, giving us the tools to calculate probabilities effectively.

Conditional Probability

Conditional probability explores the likelihood of an event occurring, given that another event has already occurred. In probability theory, it helps refine our prediction based on new information.
In our scenario, we are interested in the probability of the first four people not knowing the answer, given that only three out of the entire group know the answer. Here's how we apply it:

• Recognize the condition: The first four individuals will be chosen from those who don’t have the answer.
• Use the known counts: We have a subset of people who do not know the answer (\( n-3 \) people), and we select our first four from this group.

The conditional probability is calculated as a ratio of the favorable outcomes (first four don't know) to the total possible outcomes (any four chosen), which is expressed mathematically as:
\[ P(A | B) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]
This form of computation ensures that we focus only on the relevant probabilities, refining our overall understanding and providing a precise answer.