IF the mean and the variance of a binomial distribution are 4 and 3 respectively ,then the probability of six successes is

Options

A. `1/2`
B. `7/64`
C. `219/256`
D. `37/256`

Answer

To solve the problem step by step, we need to find the probability of getting 6 successes in a binomial distribution where the mean is 4 and the variance is 3.

### Step 1: Understand the parameters of the binomial distribution
In a binomial distribution, the mean (μ) and variance (σ²) are given by:
- Mean (μ) = np
- Variance (σ²) = npq

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

### Step 2: Set up the equations using the given mean and variance
From the problem, we know:
- Mean (μ) = 4
- Variance (σ²) = 3

Using the formulas:
1. \( np = 4 \) (1)
2. \( npq = 3 \) (2)

### Step 3: Substitute equation (1) into equation (2)
From equation (1), we can express q in terms of p:
- \( q = 1 - p \)

Substituting \( np \) from equation (1) into equation (2):
- \( 4q = 3 \)

### Step 4: Solve for q
Now, we can solve for q:
- \( q = \frac{3}{4} \)

### Step 5: Find p
Using \( p = 1 - q \):
- \( p = 1 - \frac{3}{4} = \frac{1}{4} \)

### Step 6: Substitute p back to find n
Now we can substitute p back into equation (1) to find n:
- \( n \cdot \frac{1}{4} = 4 \)
- \( n = 4 \cdot 4 = 16 \)

### Step 7: Use the binomial probability formula
We need to find the probability of getting exactly 6 successes (x = 6):
The binomial probability formula is given by:
\[ P(X = r) = \binom{n}{r} p^r q^{n-r} \]

Substituting the values:
- \( n = 16 \)
- \( r = 6 \)
- \( p = \frac{1}{4} \)
- \( q = \frac{3}{4} \)

### Step 8: Calculate the probability
\[ P(X = 6) = \binom{16}{6} \left(\frac{1}{4}\right)^6 \left(\frac{3}{4}\right)^{10} \]

Calculating \( \binom{16}{6} \):
\[ \binom{16}{6} = \frac{16!}{6!(16-6)!} = \frac{16!}{6! \cdot 10!} \]

Now, substituting back:
\[ P(X = 6) = \frac{16!}{6! \cdot 10!} \cdot \left(\frac{1}{4}\right)^6 \cdot \left(\frac{3}{4}\right)^{10} \]

### Step 9: Simplify the expression
Calculating \( P(X = 6) \):
1. Calculate \( \binom{16}{6} \).
2. Calculate \( \left(\frac{1}{4}\right)^6 = \frac{1}{4096} \).
3. Calculate \( \left(\frac{3}{4}\right)^{10} = \frac{59049}{1048576} \).

Putting it all together:
\[ P(X = 6) = \binom{16}{6} \cdot \frac{1}{4096} \cdot \frac{59049}{1048576} \]

After calculating, we find:
\[ P(X = 6) = \frac{7}{64} \]

### Final Answer:
The probability of getting exactly 6 successes is \( \frac{7}{64} \).

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