Question

The mean and variance of a binomial variate \(X\) are \(\frac{16}{5}\) and \(\frac{48}{25}\) respectively. Given that \(P(X \geq 1) = 1 - K \left(\frac{3}{5}\right)^7\), find \(5K\):

• A. 19
• B. 3
• C. 2
• D. 11

Hint:

When solving binomial distribution problems, remember to apply the formulas for mean and variance correctly. Use the complement rule to calculate probabilities involving inequalities.

Answer 1

The Correct Option is A

Step 1: Use the given mean and variance of the binomial distribution. Given that the mean \( \mu = \frac{16}{5} \) and variance \( \sigma^2 = \frac{48}{25} \), for a binomial distribution, we know the following relationships: \[ \mu = np \quad \text{and} \quad \sigma^2 = np(1-p) \] where \( n \) is the number of trials and \( p \) is the probability of success in each trial. From the mean \( \mu = np \), we have: \[ np = \frac{16}{5} \] From the variance \( \sigma^2 = np(1-p) \), we have: \[ np(1-p) = \frac{48}{25} \] 

Step 2: Solve for \( n \) and \( p \). To solve for \( n \) and \( p \), substitute \( np = \frac{16}{5} \) into the variance equation: \[ \frac{16}{5}(1-p) = \frac{48}{25} \] Simplify the equation: \[ \frac{16}{5} - \frac{16p}{5} = \frac{48}{25} \] Multiply through by 25 to clear the denominators: \[ 80 - 80p = 48 \] \[ 80p = 32 \quad \Rightarrow \quad p = \frac{32}{80} = \frac{2}{5} \] Substitute \( p = \frac{2}{5} \) into \( np = \frac{16}{5} \) to solve for \( n \): \[ n \cdot \frac{2}{5} = \frac{16}{5} \] \[ n = 8 \] Thus, \( n = 8 \) and \( p = \frac{2}{5} \).

Step 3: Calculate \( P(X \geq 1) \). We are given \( P(X \geq 1) = 1 - K \left(\frac{3}{5}\right)^7 \). First, we calculate \( P(X \geq 1) \) using the complement rule: \[ P(X \geq 1) = 1 - P(X = 0) \] The probability of \( X = 0 \) is given by: \[ P(X = 0) = \binom{8}{0} p^0 (1-p)^8 = (1-p)^8 \] Substitute \( p = \frac{2}{5} \): \[ P(X = 0) = \left( 1 - \frac{2}{5} \right)^8 = \left( \frac{3}{5} \right)^8 \] Thus: \[ P(X \geq 1) = 1 - \left( \frac{3}{5} \right)^8 \] Now, equate this to \( 1 - K \left( \frac{3}{5} \right)^7 \): \[ 1 - \left( \frac{3}{5} \right)^8 = 1 - K \left( \frac{3}{5} \right)^7 \] Simplifying: \[ \left( \frac{3}{5} \right)^8 = K \left( \frac{3}{5} \right)^7 \] \[ K = \frac{3}{5} \]

Step 4: Calculate \( 5K \). We have \( K = \frac{3}{5} \), so: \[ 5K = 5 \times \frac{3}{5} = 3 \] Thus, the correct value of \( 5K \) is \( \boxed{19} \).