Problem 168 - A Complete Resource Book in Mathematics for JEE Main

Question

Three numbers are chosen at random without replacement from first eight natural numbers. The probability that their minimum is 3 , given that their maximum is 6, is (A) \(\frac{3}{8}\) (B) \(\frac{1}{5}\) (C) \(\frac{1}{4}\) (D) \(\frac{2}{5}\)

Step-by-Step Solution

Answer

The probability is \( \frac{3}{20} \), which is not listed; thus, none of the options is correct.

Step 1: Identify Possible Numbers

Since we are choosing numbers such that their maximum is 6, the possible numbers we can pick are 1, 2, 3, 4, 5, and 6.

Step 2: Count Total Combinations

We need to find all combinations of choosing three numbers from the set \{1, 2, 3, 4, 5, 6\}. The total number of combinations is given by the combination formula \( \binom{n}{r} \), where \( n \) is the total number of elements to choose from, and \( r \) is the number of elements to choose. This gives us \( \binom{6}{3} = 20 \).

Step 3: Count Favorable Combinations

The favorable outcomes are those where the minimum is 3. If 3 is the minimum, all numbers must be either 3, 4, 5, or 6. We need to choose 3 numbers from this subset where 3 is included. So the other two numbers can be chosen from \{4, 5, 6\}. This can be done in \( \binom{3}{2} = 3 \) ways.

Step 4: Apply Conditional Probability Formula

The conditional probability formula is \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( B \) is the event "maximum is 6", and \( A \) is the event "minimum is 3". We calculated \( P(A \cap B) = 3 \) from favorable outcomes and \( P(B) = 20 \) as the total combinations. So, \( P(A|B) = \frac{3}{20} \).

Step 5: Calculate Simplified Probability

Simplify the fraction obtained: \( \frac{3}{20} \). This cannot be simplified further, so \( \frac{3}{20} \) is the probability that their minimum is 3, given that their maximum is 6.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics that deals with counting, arrangement, and combination of objects. In problems involving combinatorics, we often ask the "how many?" questions.
In the given exercise, we use combinatorics to find the number of ways to choose certain numbers from a set of natural numbers. Specifically, if you want to find out the total number of ways to choose 3 numbers from a set of 6 numbers, you would use the combination formula:

• The combination formula is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items to select.
• Here, \( n = 6 \) and \( r = 3 \), which gives us \( \binom{6}{3} = 20 \) ways.

Combinatorics is essential in probability studies because it helps in determining the total possible outcomes, a critical factor when calculating probabilities.

Conditional Probability

Conditional probability is used to determine the likelihood of an event occurring, given that another event has already occurred. It is expressed with the formula:

• \( P(A|B) = \frac{P(A \cap B)}{P(B)} \), where \( A \) and \( B \) are two events, \( P(A \cap B) \) is the likelihood both \( A \) and \( B \) occur, and \( P(B) \) is the probability of event \( B \) occurring.

In our problem, event \( B \) refers to the maximum number being 6, and event \( A \) relates to the minimum number being 3. First, we counted all combinations leading to the maximum being 6 (20 combinations).
Next, we identified the number of cases where both the maximum is 6 and minimum is 3 (3 combinations), which is our \( P(A \cap B) \).
Finally, the conditional probability \( P(A|B) \) is found to be \( \frac{3}{20} \), providing the probability that the smallest selected number is 3, given the largest is 6.

Natural Numbers

Natural numbers are a sequence of positive integers starting from 1, which are used for counting. In mathematics, they are typically represented by the symbol \( \mathbb{N} \) and include numbers like 1, 2, 3, 4, and so on.
In the context of the problem, the first eight natural numbers are considered: 1, 2, 3, 4, 5, 6, 7, and 8. However, since the maximum allowed number is 6, only numbers 1 through 6 are relevant to our scenario.
Understanding natural numbers is fundamental because they form the first step in solving many mathematical problems. They provide a simple framework for discussing and solving basic mathematical operations like addition, subtraction, and more complex topics like the one explored in this exercise.