The mean and standard deviation of a binomial variate X are 4 and `sqrt(3)` respectively. Then ,`P(Xge1)=`

Options

A. `1-((1)/(4))^16`
B. `1-((3)/(4))^16`
C. `1-((2)/(3))^16`
D. `1-((1)/(3))^16`

Answer

To solve the problem, we need to find \( P(X \geq 1) \) for a binomial random variable \( X \) with given mean and standard deviation. Let's go through the solution step by step.

### Step 1: Understand the parameters of the binomial distribution
The mean \( \mu \) and standard deviation \( \sigma \) of a binomial distribution can be expressed as:
- Mean: \( \mu = np \)
- Standard Deviation: \( \sigma = \sqrt{np(1-p)} \)

Where:
- \( n \) = number of trials
- \( p \) = probability of success
- \( q \) = probability of failure = \( 1 - p \)

### Step 2: Set up equations based on given values
From the problem, we know:
- Mean \( \mu = 4 \)
- Standard deviation \( \sigma = \sqrt{3} \)

Using these, we can set up the following equations:
1. \( np = 4 \) (1)
2. \( \sqrt{np(1-p)} = \sqrt{3} \) (2)

### Step 3: Square the standard deviation equation
From equation (2):
\[ np(1-p) = 3 \]
Substituting \( np = 4 \) from equation (1) into this equation:
\[ 4(1-p) = 3 \]

### Step 4: Solve for \( p \)
Rearranging the equation:
\[ 1 - p = \frac{3}{4} \]
Thus,
\[ p = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 5: Find \( n \)
Now substitute \( p \) back into equation (1):
\[ n \cdot \frac{1}{4} = 4 \]
Multiplying both sides by 4:
\[ n = 16 \]

### Step 6: Calculate \( P(X \geq 1) \)
To find \( P(X \geq 1) \), we use the complement rule:
\[ P(X \geq 1) = 1 - P(X < 1) = 1 - P(X = 0) \]

### Step 7: Calculate \( P(X = 0) \)
Using the binomial probability formula:
\[ P(X = 0) = \binom{n}{0} p^0 q^n = 1 \cdot (1/4)^0 \cdot (3/4)^{16} \]
This simplifies to:
\[ P(X = 0) = \left(\frac{3}{4}\right)^{16} \]

### Step 8: Final calculation for \( P(X \geq 1) \)
Now substituting back:
\[ P(X \geq 1) = 1 - \left(\frac{3}{4}\right)^{16} \]

### Conclusion
Thus, the final answer is:
\[ P(X \geq 1) = 1 - \left(\frac{3}{4}\right)^{16} \]