If the mean and standard deviation of a binomial distribution are 12 and 2 respectively , then the value of its parameter p is

Options

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

Answer

To solve the problem step by step, we will use the properties of a binomial distribution.

### Step 1: Understand the Mean and Standard Deviation of a Binomial Distribution
The mean (μ) and standard deviation (σ) of a binomial distribution can be expressed as:
- Mean (μ) = np
- Standard Deviation (σ) = √(npq)

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

### Step 2: Set Up the Equations
From the problem, we know:
- Mean (μ) = 12
- Standard Deviation (σ) = 2

Using the formulas:
1. np = 12 (Equation 1)
2. σ = √(npq) = 2 (Equation 2)

### Step 3: Square the Standard Deviation Equation
From Equation 2, we square both sides:
(σ)^2 = npq
2^2 = npq
4 = npq (Equation 3)

### Step 4: Substitute np from Equation 1 into Equation 3
From Equation 1, we have np = 12. We can substitute this into Equation 3:
4 = 12q

### Step 5: Solve for q
Now, we solve for q:
4 = 12q
=> q = 4/12
=> q = 1/3

### Step 6: Find p
We know that p + q = 1. Thus:
p + 1/3 = 1
=> p = 1 - 1/3
=> p = 2/3

### Conclusion
The value of the parameter p is **2/3**.