If mean and variance of a Binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than 1 is:

Options

A. A) `(2)/(3)`
B. B) `(4)/(5)`
C. C) `(7)/(8)`
D. D) `(11)/(16)`

Answer

To solve the problem step by step, we will follow the information provided in the question regarding the mean and variance of the binomial distribution.

### Step 1: Understand the Mean and Variance of a Binomial Distribution
For a binomial distribution:
- 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) = 2
- Variance (σ²) = 1

### Step 2: Set Up the Equations
From the given information, we can set up the following equations:
1. \( np = 2 \) (Equation 1)
2. \( npq = 1 \) (Equation 2)

### Step 3: Express q in Terms of p
From Equation 1, we can express q as follows:
- Since \( q = 1 - p \), we can substitute this into Equation 2.

### Step 4: Substitute q into the Variance Equation
Substituting \( q = 1 - p \) into Equation 2:
\[ np(1 - p) = 1 \]
Using Equation 1 (where \( np = 2 \)):
\[ 2(1 - p) = 1 \]
Solving for p:
\[ 2 - 2p = 1 \implies 2p = 1 \implies p = \frac{1}{2} \]

### Step 5: Find q
Now that we have p, we can find q:
\[ q = 1 - p = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 6: Find n
Using the value of p in Equation 1:
\[ n \cdot \frac{1}{2} = 2 \implies n = 2 \cdot 2 = 4 \]

### Step 7: Find the Probability that X > 1
We need to find \( P(X > 1) \). This can be calculated as:
\[ P(X > 1) = 1 - P(X \leq 1) = 1 - (P(X = 0) + P(X = 1)) \]

### Step 8: Calculate P(X = 0) and P(X = 1)
Using the binomial probability formula:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]

1. For \( P(X = 0) \):
\[ P(X = 0) = \binom{4}{0} \left(\frac{1}{2}\right)^0 \left(\frac{1}{2}\right)^{4} = 1 \cdot 1 \cdot \frac{1}{16} = \frac{1}{16} \]

2. For \( P(X = 1) \):
\[ P(X = 1) = \binom{4}{1} \left(\frac{1}{2}\right)^1 \left(\frac{1}{2}\right)^{3} = 4 \cdot \frac{1}{2} \cdot \frac{1}{8} = \frac{4}{16} = \frac{1}{4} \]

### Step 9: Combine the Probabilities
Now, we can find \( P(X \leq 1) \):
\[ P(X \leq 1) = P(X = 0) + P(X = 1) = \frac{1}{16} + \frac{4}{16} = \frac{5}{16} \]

### Step 10: Calculate P(X > 1)
Finally, we can find \( P(X > 1) \):
\[ P(X > 1) = 1 - P(X \leq 1) = 1 - \frac{5}{16} = \frac{11}{16} \]

### Final Answer
The probability that \( X \) takes a value greater than 1 is \( \frac{11}{16} \).

---