In binomial probability distribution, mean is 3 and standard deviation is `3/2`. Then the value of p is

Options

A. `1/4`
B. `3/4`
C. `1/2`
D. `1/3`

Answer

To find the value of \( p \) in a binomial probability distribution where the mean is 3 and the standard deviation is \( \frac{3}{2} \), we can follow these steps:

### Step 1: Understand the formulas
In a binomial distribution:
- The mean \( \mu \) is given by \( \mu = np \)
- The variance \( \sigma^2 \) is given by \( \sigma^2 = npq \)
- The standard deviation \( \sigma \) is the square root of the variance, \( \sigma = \sqrt{npq} \)

### Step 2: Write down the given information
We know:
- Mean \( \mu = 3 \)
- Standard deviation \( \sigma = \frac{3}{2} \)

### Step 3: Set up the equations
From the mean:
\[ np = 3 \quad \text{(Equation 1)} \]
From the standard deviation:
\[ \sigma^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \]
Thus, we have:
\[ npq = \frac{9}{4} \quad \text{(Equation 2)} \]

### Step 4: Substitute Equation 1 into Equation 2
From Equation 1, we can express \( q \) in terms of \( p \):
\[ q = 1 - p \]
Substituting \( np = 3 \) into Equation 2:
\[ 3q = \frac{9}{4} \]
Now substituting \( q = 1 - p \):
\[ 3(1 - p) = \frac{9}{4} \]

### Step 5: Solve for \( p \)
Expanding the equation:
\[ 3 - 3p = \frac{9}{4} \]
Rearranging gives:
\[ 3p = 3 - \frac{9}{4} \]
To combine the terms on the right, convert 3 into quarters:
\[ 3 = \frac{12}{4} \]
Thus:
\[ 3p = \frac{12}{4} - \frac{9}{4} = \frac{3}{4} \]
Now, divide both sides by 3:
\[ p = \frac{3}{4} \cdot \frac{1}{3} = \frac{1}{4} \]

### Conclusion
The value of \( p \) is:
\[ \boxed{\frac{1}{4}} \]