Question

In a Binomial distribution, the difference between the mean and standard deviation is 3, and the difference between their squares is 21. Then, the ratio \( P(x = 1) : P(x = 2) \) is:

• A. \( 2 : 1 \)
• B. \( 1 : 2 \)
• C. \( 1 : 3 \)
• D. \( 3 : 1 \)

Hint:

In binomial probability problems, forming equations using mean and variance properties helps in solving the problem step by step.

Answer 1

The Correct Option is C

Step 1: Define the Binomial Distribution A binomial distribution is given by: \[ X \sim B(n, p) \] where: - \( n \) is the number of trials, - \( p \) is the probability of success. The mean and standard deviation of a binomial distribution are: \[ \text{Mean} = \mu = np \] \[ \text{Standard Deviation} = \sigma = \sqrt{np(1 - p)} \]

Step 2: Given Conditions and Forming Equations We are given: \[ \mu - \sigma = 3 \] \[ \mu^2 - \sigma^2 = 21 \] Substituting \( \mu = np \) and \( \sigma^2 = np(1 - p) \): \[ np - \sqrt{np(1 - p)} = 3 \] Squaring both sides: \[ % Option (np)^2 - np(1 - p) = 21 \]
Step 3: Solve for \( n \) and \( p \) Let \( x = np \), so we rewrite the equations: \[ x - \sqrt{x(1 - p)} = 3 \] Squaring both sides: \[ x^2 - x(1 - p) = 21 \] Rewriting \( x(1 - p) \) from the first equation: \[ (x - 3)^2 = x(1 - p) \] Substituting in the second equation: \[ x^2 - (x - 3)^2 = 21 \] Expanding: \[ x^2 - (x^2 - 6x + 9) = 21 \] \[ x^2 - x^2 + 6x - 9 = 21 \] \[ 6x = 30 \] \[ x = 5 \] Since \( x = np = 5 \), substituting back into the first equation: \[ 5 - \sqrt{5(1 - p)} = 3 \] \[ \sqrt{5(1 - p)} = 2 \] Squaring: \[ 5(1 - p) = 4 \] \[ 1 - p = \frac{4}{5} \] \[ p = \frac{1}{5} \] Since \( np = 5 \), we find \( n \): \[ n \cdot \frac{1}{5} = 5 \] \[ n = 25 \]
Step 4: Compute Probability Ratio The binomial probability formula is: \[ P(x = k) = \binom{n}{k} p^k (1 - p)^{n - k} \] For \( P(x = 1) \): \[ P(x = 1) = \binom{25}{1} \left(\frac{1}{5}\right)^1 \left(\frac{4}{5}\right)^{24} \] For \( P(x = 2) \): \[ P(x = 2) = \binom{25}{2} \left(\frac{1}{5}\right)^2 \left(\frac{4}{5}\right)^{23} \] Taking their ratio: \[ \frac{P(x = 1)}{P(x = 2)} = \frac{\binom{25}{1} \cdot \frac{1}{5} \cdot \left(\frac{4}{5}\right)^{24}}{\binom{25}{2} \cdot \frac{1}{5^2} \cdot \left(\frac{4}{5}\right)^{23}} \] \[ = \frac{25 \times \frac{1}{5}}{\frac{25 \times 24}{2} \times \frac{1}{25}} \] \[ = \frac{5}{\frac{600}{2}} \] \[ = \frac{5}{300} \] \[ = \frac{1}{3} \] Thus: \[ P(x = 1) : P(x = 2) = 1 : 3 \]