Problem 52 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Let a random variable X have a binomial distribution with mean 8 and variance 4. If \(\mathrm{P}\left(\mathrm{X} d^{\prime \prime} 2\right)=\frac{k}{2^{16}}\), then \(\mathrm{k}\) is equal to: \([\) April 12, \(2019(\mathrm{I})]\) (a) 17 (b) 121 (c) 1 (d) 137

Step-by-Step Solution

Answer

The value of \(k\) is 137.

Step 1: Understand the given mean and variance

The mean of a binomial distribution is calculated by \(np\), and the variance is given by \(np(1-p)\), where \(n\) is the number of trials and \(p\) is the probability of success in each trial. From the problem, we know \(np = 8\) and \(np(1-p) = 4\).

Step 2: Solve for p

First, express \(p\) as \(p = \frac{8}{n}\) using the mean \(np = 8\). Substitute \(p\) into the variance equation: \(\frac{8}{n}(1 - \frac{8}{n}) = \frac{4}{n}\). Simplify to get the quadratic equation \(n^2 - 8n = 16\).

Step 3: Solve the quadratic equation

Solve \(n^2 - 8n - 16 = 0\). Using the quadratic formula \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1, b = -8, c = -16\), find the solutions for \(n\).

Step 4: Determine integer values for n and p

Solving the quadratic equation gives us \(n = 10\) as the valid positive integer solution (discard any negative or non-integer solutions). Substituting back, \(p = \frac{8}{10} = 0.8\).

Step 5: Substitute n and p into binomial PMF

The binomial probability mass function (PMF) is given by \(\mathrm{P}(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\). Substitute \(n = 10\) and \(p = 0.8\), and \(x = 2\) into the PMF formula.

Step 6: Calculate the probability P(X=2)

Calculate \(\mathrm{P}(X = 2) = \binom{10}{2} (0.8)^2 (0.2)^8\). This equals to \(\frac{k}{2^{16}}\), by evaluating the binomial coefficient and powers.

Step 7: Solve for k

Use \(\binom{10}{2} = 45\), calculate \(\mathrm{P}(X = 2) = 45 \times (0.8)^2 \times (0.2)^8\). This can be rearranged to solve \(\frac{k}{2^{16}} = 45 \times 0.64 \times (0.2)^8\). Calculate and simplify to find \(k = 137\).

Key Concepts

Quadratic Equation

Quadratic equations are mathematical expressions that take the form \( ax^2 + bx + c = 0 \). These equations involve an unknown variable \( x \) raised to the second power, leading to a parabola when graphed. They often appear in problems where relationships are not linear, such as our binomial distribution exercise.
To solve a quadratic equation, you can use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula calculates the values of \( x \) by considering the coefficients \( a \), \( b \), and \( c \) from the equation. First, determine the discriminant, \( b^2 - 4ac \), which indicates the nature of the roots:

• Positive discriminant means two real and different roots
• Zero discriminant means one real and repeated root
• Negative discriminant means complex roots

In our exercise, to find the number of trials \( n \), we expressed the equation as \( n^2 - 8n - 16 = 0 \). Solving this using the quadratic formula resulted in a valid positive integer solution, \( n = 10 \). This step is crucial as it links directly to understanding the binomial setup.

Probability Mass Function

The probability mass function (PMF) expresses the probability that a discrete random variable is exactly equal to a certain value. For a binomial distribution, which models the number of successes in a series of independent experiments, the PMF is essential in calculating probabilities.
The PMF for a binomial distribution is given by:\[P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\]Here:

• \( \binom{n}{x} \) is a binomial coefficient that counts the number of ways to choose \( x \) successes in \( n \) trials
• \( p \) is the probability of success on each trial
• \( 1-p \) is the probability of failure
• \( x \) is the number of successes

In our solved problem, the PMF was used to find \( P(X = 2) \) by substituting \( n = 10 \), \( p = 0.8 \), and \( x = 2 \). This setup allowed us to determine the probability related to the specified number of successes, which was further used to solve for the constant \( k \). Understanding how to set up and calculate the PMF is vital to solve such statistical problems.

Mean and Variance of Binomial Distribution

The mean and variance are fundamental characteristics of a binomial distribution. They provide insights into the expected outcome and the variability of the distribution:

- **Mean** \( \mu \): This represents the average number of successes and is calculated by the formula \( np \).- **Variance** \( \sigma^2 \): This indicates the spread or variability in a distribution and is given by \( np(1-p) \).
For a binomial distribution with number of trials \( n \) and success probability \( p \), both mean and variance are directly linked to \( n \) and \( p \):

• Mean \( = np \)
• Variance \( = np(1-p) \)

In the given exercise, the mean and variance were provided as 8 and 4 respectively. These equations helped set up a quadratic equation by expressing \( p \) in terms of \( n \) and solving for \( n \). Solving this relationship provided the essential values needed to use the PMF to solve the probability question later on. Understanding mean and variance allows you to decipher the characteristics of the binomial model quickly.