The mean and variance of a binomial distribution are 4 and 3 respectively. Fix the distribution and find `P(X ge 1)`.

Answer

To solve the problem, we need to find the parameters of a binomial distribution given the mean and variance, and then calculate the probability \( P(X \geq 1) \).

### Step-by-Step Solution:

1. **Identify the Mean and Variance Formulas**:
The mean (\( \mu \)) and variance (\( \sigma^2 \)) of a binomial distribution are given by:
\[ \mu = nP \quad \text{(1)} \]
\[ \sigma^2 = nPQ \quad \text{(2)} \]
where \( n \) is the number of trials, \( P \) is the probability of success, and \( Q = 1 - P \) is the probability of failure.

2. **Substitute the Given Values**:
From the problem, we have:
\[ \mu = 4 \quad \text{and} \quad \sigma^2 = 3 \]
Thus, we can write:
\[ nP = 4 \quad \text{(from 1)} \]
\[ nPQ = 3 \quad \text{(from 2)} \]

3. **Express \( Q \) in Terms of \( P \)**:
Since \( Q = 1 - P \), we can substitute this into equation (2):
\[ nP(1 - P) = 3 \quad \text{(3)} \]

4. **Divide Equation (3) by Equation (1)**:
Dividing equation (3) by equation (1) gives:
\[ \frac{nP(1 - P)}{nP} = \frac{3}{4} \]
Simplifying this, we find:
\[ 1 - P = \frac{3}{4} \]
Therefore, solving for \( P \):
\[ P = 1 - \frac{3}{4} = \frac{1}{4} \]
Now substituting \( P \) back into equation (1):
\[ n \cdot \frac{1}{4} = 4 \implies n = 16 \]

5. **Find \( Q \)**:
Now we can find \( Q \):
\[ Q = 1 - P = 1 - \frac{1}{4} = \frac{3}{4} \]

6. **Calculate \( P(X \geq 1) \)**:
We need to find \( P(X \geq 1) \). This can be calculated using the complement rule:
\[ P(X \geq 1) = 1 - P(X = 0) \]
The probability \( P(X = 0) \) is given by:
\[ P(X = 0) = \binom{n}{0} P^0 Q^n = \binom{16}{0} \left(\frac{1}{4}\right)^0 \left(\frac{3}{4}\right)^{16} \]
Since \( \binom{16}{0} = 1 \) and \( \left(\frac{1}{4}\right)^0 = 1 \):
\[ P(X = 0) = 1 \cdot 1 \cdot \left(\frac{3}{4}\right)^{16} \]
Thus:
\[ P(X = 0) = \left(\frac{3}{4}\right)^{16} \]

7. **Final Calculation**:
Now substituting back into the equation for \( P(X \geq 1) \):
\[ P(X \geq 1) = 1 - \left(\frac{3}{4}\right)^{16} \]

### Final Answer:
\[ P(X \geq 1) = 1 - \left(\frac{3}{4}\right)^{16} \]