The mean and variance of a binomial distribution are 8 and 4 respectively. What is P(X = 1) equal to ?

Options

A. `1/2^(12)`
B. `1/2^(8)`
C. `1/2^(6)`
D. `1/2^(4)`

Explanation

To solve the problem step by step, we will use the properties of the binomial distribution. We know the mean (μ) and variance (σ²) of a binomial distribution are given by the formulas:

- Mean (μ) = NP
- Variance (σ²) = NPQ

where:
- N = number of trials
- P = probability of success
- Q = probability of failure (Q = 1 - P)

Given:
- Mean (μ) = 8
- Variance (σ²) = 4

### Step 1: Set up the equations for mean and variance
From the information provided, we can set up the following equations:
1. NP = 8 (1)
2. NPQ = 4 (2)

### Step 2: Express Q in terms of P
From equation (1), we can express Q as:
- Q = 1 - P

### Step 3: Substitute Q in the variance equation
Substituting Q in equation (2):
- NP(1 - P) = 4
- NP - NP² = 4

### Step 4: Substitute NP from the mean equation
From equation (1), we know NP = 8. Substitute this into the equation:
- 8 - 8P = 4

### Step 5: Solve for P
Rearranging gives:
- 8P = 8 - 4
- 8P = 4
- P = 4/8 = 1/2

### Step 6: Find Q
Now that we have P, we can find Q:
- Q = 1 - P = 1 - 1/2 = 1/2

### Step 7: Find N using the mean
Substituting P back into equation (1):
- NP = 8
- N(1/2) = 8
- N = 8 * 2 = 16

### Step 8: Calculate P(X = 1)
Now we need to find P(X = 1) using the binomial probability formula:
\[ P(X = k) = \binom{N}{k} P^k Q^{N-k} \]

For k = 1:
\[ P(X = 1) = \binom{16}{1} (1/2)^1 (1/2)^{16-1} \]
\[ P(X = 1) = 16 \cdot (1/2) \cdot (1/2)^{15} \]
\[ P(X = 1) = 16 \cdot (1/2)^{16} \]
\[ P(X = 1) = 16 \cdot \frac{1}{2^{16}} \]
\[ P(X = 1) = \frac{16}{2^{16}} \]
\[ P(X = 1) = \frac{1}{2^{12}} \]

### Final Answer
Thus, \( P(X = 1) = \frac{1}{2^{12}} \).