Problem 66 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Three numbers are chosen at random without replacement from \(\\{1,2,3, \ldots, 8]\). The probability that their minimum is 3 , given that their maximum is 6 , is: (A) \(3 / 8\) (B) \(1 / 5\) (C) \(1 / 4\) (D) \(2 / 5\)

Step-by-Step Solution

Answer

The probability is (B) \(1/5\).

Step 1: Understanding the Conditional Probability

We are asked to find the probability that the minimum of the chosen three numbers is 3, given that the maximum is 6. This is represented as \( P(\text{min} = 3 \mid \text{max} = 6) \).

Step 2: Determine Total Ways to Select Numbers with Max 6

First, calculate the total number of ways to select three numbers from \( \{1, 2, 3, 4, 5, 6\} \), as this is the condition for the maximum being 6. The number of ways is given by \( \binom{6}{3} \).

Step 3: Using Combination Formula

Calculate \( \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20 \). There are 20 ways to pick any 3 numbers from \{1, 2, 3, 4, 5, 6\}.

Step 4: Determine Ways for Minimum to be 3

For the minimum to be 3, we must select 3 and choose the other two numbers from \{3, 4, 5, 6\}. So we choose 2 additional numbers from \{4, 5, 6\}, calculated with \( \binom{3}{2} \).

Step 5: Using Combination Formula for Minimum Condition

Calculate \( \binom{3}{2} = \frac{3 \times 2}{2 \times 1} = 3 \). Hence, there are 3 ways to choose these numbers.

Step 6: Calculate Conditional Probability

The probability that the minimum is 3 given the maximum is 6 is \( \frac{\text{number of favorable outcomes}}{\text{total outcomes with max 6}} = \frac{3}{20} \).

Step 7: Convert Probability to Simplest Fraction

Thus, the probability simplifies to \( 1/5 \).

Key Concepts

Combinatorics

Combinatorics is an essential branch of mathematics concerned with counting, arrangement, and combination of objects. It helps solve problems related to probability by allowing us to count the number of possible ways an event can occur.

The exercise requires selecting three numbers from a set, starting with the set \(\{1, 2, 3, 4, 5, 6, 7, 8\}\). When dealing with such selections where the order does not matter, combinatorics simplifies the process by using combinations.

Combinations tell us how many ways we can select a specific number of objects from a larger set. This is useful in scenarios like our exercise, where we need to count the selection of numbers under certain conditions.

Binomial Coefficient

Binomial coefficients are at the heart of calculating combinations. The binomial coefficient \(\binom{n}{r}\) tells us how many ways we can choose \(r\) objects from a total of \(n\) objects without considering the order. This is also known as "n choose r".

In our problem, the solution involves choosing 3 numbers from the first 6 numbers, \(\{1, 2, 3, 4, 5, 6\}\), while ensuring the maximum number is 6. This selection is calculated as \(\binom{6}{3}\).

The formula for calculating a binomial coefficient is given by:

• \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\)

This expression gives the number of ways to combine items from a set, which is crucial in determining the number of possible outcomes for our conditions.

Each step in solving the probability problem involves assessing such combinations, making binomial coefficients a foundational tool.

Probability Calculation

Probability calculation involves determining the likelihood of a particular outcome from all possible outcomes. In our exercise, this is calculated using conditional probability, where we determine the likelihood of one event given that another event has already occurred.

The conditional probability is symbolized as \(P(A \mid B)\), which reads as the probability of event \(A\) occurring, provided event \(B\) has occurred. Here, \(A\) is the minimum being 3, and \(B\) is the maximum being 6.

Steps to calculate it:

• Determine the total number of ways to achieve the condition \(B\) (maximum is 6).
• Count the number of ways both \(A\) and \(B\) can happen, which is when the minimum is 3, and the maximum is 6.
• The conditional probability is the ratio of the number of favorable outcomes to the total outcomes under the condition \(B\).

Using these steps, we get \(\frac{3}{20}\) for our probability fraction within the exercise, which further simplifies to \(\frac{1}{5}\).

Such a probability calculation is crucial for understanding how different chances play out under specific conditions.