Problem 46 - A Complete Resource Book in Mathematics for JEE Main

Question

In a binomial distribution \(B\left(n, p=\frac{1}{4}\right)\), if the probability of at least one success is greater than equal to \(\frac{9}{10}\), then \(n\) is greater than (A) \(\frac{1}{\log _{10}{\underline{\phantom{xx}}}^{4}-\log _{10}{\underline{\phantom{xx}}}^{3}}\) (B) \(\frac{1}{\log _{10}{\underline{\phantom{xx}}}^{4}+\log _{10}{\underline{\phantom{xx}}}^{3}}\) (C) \(\frac{9}{\log _{10}^{4}-\log _{10}{\underline{\phantom{xx}}}^{3}}\) (D) \(\frac{4}{\log _{10}{\underline{\phantom{xx}}}^{4}-\log _{10}{\underline{\phantom{xx}}}^{3}}\)

Step-by-Step Solution

Answer

Option (A): \( \frac{1}{\log_{10}{4} - \log_{10}{3}} \).

Step 1: Understanding the Problem

The problem involves a binomial distribution where the probability of success is \( p = \frac{1}{4} \). We need to find the value of \( n \) such that the probability of getting at least one success is at least \( \frac{9}{10} \).

Step 2: Using Complement Rule

We know that the probability of at least one success is \( 1 - P(\text{no success}) \). If \( X \) is the random variable for this binomial distribution, \( P(\text{no success}) = (1-p)^n = \left(\frac{3}{4}\right)^n \).

Step 3: Set Up Inequality

Since we want the probability of at least one success to be greater than or equal to \( \frac{9}{10} \), we set up the inequality: \[ 1 - \left(\frac{3}{4}\right)^n \geq \frac{9}{10} \].

Step 4: Solve for \( n \)

Rewriting the inequality, we get: \[ \left(\frac{3}{4}\right)^n \leq \frac{1}{10} \]. Taking logarithm on both sides yields: \[ n \cdot \log_{10}\left(\frac{3}{4}\right) \leq \log_{10}\left(\frac{1}{10}\right) \].

Step 5: Simplify with Logarithms

Rearranging the inequality gives us:\[ n \geq \frac{\log_{10}\left(\frac{1}{10}\right)}{\log_{10}\left(\frac{4}{3}\right)} \]. Simplifying, we have:\[ n \geq \frac{-1}{\log_{10}\left(\frac{4}{3}\right)} \].

Step 6: Evaluate Options

Given options suggest transformations relate to this expression. The transformation involves using laws of logarithms to consider potential expressions involving \( \log_{10}{4} \) and \( \log_{10}{3} \) separately. Ultimately, it relates to \[ n > \frac{1}{\log_{10} 4 - \log_{10} 3} \].

Step 7: Choose the Correct Option

The calculation for the inequality results in:\[ n > \frac{1}{\log_{10} 4 - \log_{10} 3} \]. This matches option (A).Therefore, option (A) is correct: \( \frac{1}{\log_{10}{4} - \log_{10}{3}} \).

Key Concepts

Probability of Success

In a binomial distribution, the probability of success is a fundamental element that defines how likely an event is to occur within a given number of trials. For instance, when the problem mentions that the probability of success is \( p = \frac{1}{4} \), it implies that each trial has a 25% chance of success. This sets the stage for analyzing the distribution of successes across multiple trials.

The binomial distribution is represented by two parameters: \( n \), the number of trials, and \( p \), the probability of success in each trial. It helps to model scenarios where there are repeated independent experiments with only two possible outcomes, typically termed as "success" and "failure." As such, every trial is identical, and the outcome does not impact others, maintaining independence throughout.

In our exercise, understanding this concept allows us to establish the required conditions for the probability of at least one success, effectively guiding the setup of our inequality and calculations.

Complement Rule

The complement rule is a useful strategy in probability for calculating the probability of an event by considering the probability of its complement. In simple terms, if you know the probability of event \( A \) not happening, you can easily find the probability of it happening by subtracting from one.

In the context of this problem, the complement of achieving at least one success is achieving no success at all in all of the trials. Mathematically, this can be expressed as:

• Probability of no success \( = (1-p)^n = \left(\frac{3}{4}\right)^n \)

So, the probability of at least one success can be calculated using:

• \( 1 - \) Probability of no success \( = 1 - \left(\frac{3}{4}\right)^n \)

By applying the complement rule, the exercise guides us to set up the crucial inequality

• \( 1 - \left(\frac{3}{4}\right)^n \geq \frac{9}{10} \)

This allows for the transformation of the problem statement into a solvable mathematical expression, moving us a step closer to identifying \( n \).

Inequality with Logarithms

When setting up inequalities involving exponential terms, logarithms can greatly simplify the process. In this problem, once we have the inequality for achieving at least one success:

• \( \left(\frac{3}{4}\right)^n \leq \frac{1}{10} \)

Taking the logarithm of both sides helps isolate \( n \) and makes the numbers more manageable:

• \( n \cdot \log_{10}\left(\frac{3}{4}\right) \leq \log_{10}\left(\frac{1}{10}\right) \)

Moving forward requires understanding properties of logarithms, such as the change of base formula. The expression simplifies to exploiting the relationships:

• \( n \geq \frac{-1}{\log_{10}\left(\frac{4}{3}\right)} \)

Finally, by using rules of logarithms and identities such as \( \log(a) - \log(b) = \log\left(\frac{a}{b}\right) \), the solution identifies that:

• \( n > \frac{1}{\log_{10} 4 - \log_{10} 3} \)

This enabled the selection of the correct answer from the given options, linking what might initially seem like abstract logarithmic operations with a concrete solution to a probability problem.