In a binomial distribution n=400, `p=1//5`, It's standard deviation is

Answer

To find the standard deviation of a binomial distribution with parameters \( n = 400 \) and \( p = \frac{1}{5} \), we can follow these steps:

### Step 1: Understand the formula for standard deviation in a binomial distribution.
The standard deviation \( \sigma \) of a binomial distribution is given by the formula:
\[ \sigma = \sqrt{n \cdot p \cdot (1 - p)} \]
where \( n \) is the number of trials, \( p \) is the probability of success, and \( (1 - p) \) is the probability of failure.

### Step 2: Substitute the values into the formula.
Given:
- \( n = 400 \)
- \( p = \frac{1}{5} \)

We can substitute these values into the formula:
\[ \sigma = \sqrt{400 \cdot \frac{1}{5} \cdot (1 - \frac{1}{5})} \]

### Step 3: Calculate \( 1 - p \).
First, calculate \( 1 - p \):
\[ 1 - p = 1 - \frac{1}{5} = \frac{4}{5} \]

### Step 4: Substitute \( 1 - p \) back into the formula.
Now substitute \( 1 - p \) into the formula:
\[ \sigma = \sqrt{400 \cdot \frac{1}{5} \cdot \frac{4}{5}} \]

### Step 5: Simplify the expression inside the square root.
Now, simplify the expression:
\[ \sigma = \sqrt{400 \cdot \frac{1 \cdot 4}{5 \cdot 5}} = \sqrt{400 \cdot \frac{4}{25}} \]
\[ = \sqrt{\frac{1600}{25}} = \sqrt{64} = 8 \]

### Step 6: State the final answer.
Thus, the standard deviation \( \sigma \) is:
\[ \sigma = 8 \]

### Final Answer:
The standard deviation of the binomial distribution is **8**.

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