In a binomial distribution the mean is 12 and the standard deviation is `sqrt3`. Then the distribution is

Options

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

Answer

To solve the problem, we need to find the parameters of the binomial distribution given the mean and standard deviation. Let's break it down step by step.

### Step 1: Understand the Mean and Standard Deviation of a Binomial Distribution
The mean (μ) of a binomial distribution is given by the formula:
\[ \mu = n \cdot p \]
where \( n \) is the number of trials and \( p \) is the probability of success.

The standard deviation (σ) is given by:
\[ \sigma = \sqrt{n \cdot p \cdot q} \]
where \( q = 1 - p \) is the probability of failure.

### Step 2: Set Up the Equations
From the problem, we know:
\[ \mu = 12 \quad \text{and} \quad \sigma = \sqrt{3} \]
This gives us two equations:
1. \( n \cdot p = 12 \) (1)
2. \( n \cdot p \cdot q = 3 \) (since \( \sigma^2 = 3 \)) (2)

### Step 3: Express q in Terms of p
From equation (1), we can express \( q \) in terms of \( p \):
\[ q = 1 - p \]

### Step 4: Substitute q into the Standard Deviation Equation
Substituting \( q \) into equation (2):
\[ n \cdot p \cdot (1 - p) = 3 \]
Now substitute \( n \cdot p = 12 \) from equation (1):
\[ 12 \cdot (1 - p) = 3 \]

### Step 5: Solve for p
Now, solve for \( p \):
\[ 12 - 12p = 3 \]
\[ 12p = 12 - 3 \]
\[ 12p = 9 \]
\[ p = \frac{9}{12} = \frac{3}{4} \]

### Step 6: Find q
Now that we have \( p \), we can find \( q \):
\[ q = 1 - p = 1 - \frac{3}{4} = \frac{1}{4} \]

### Step 7: Find n
Now, substitute \( p \) back into equation (1) to find \( n \):
\[ n \cdot \frac{3}{4} = 12 \]
\[ n = 12 \cdot \frac{4}{3} = 16 \]

### Step 8: Write the Binomial Distribution
Now we have:
- \( n = 16 \)
- \( p = \frac{3}{4} \)
- \( q = \frac{1}{4} \)

The binomial distribution can be expressed as:
\[ B(n, p) = B(16, \frac{3}{4}) \]

### Final Answer
Thus, the required binomial distribution is:
\[ B(16, \frac{3}{4}) \]

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