To solve the problem of finding the value of [tex]\( n \)[/tex] in a binomial distribution where the variance is given as 1 and [tex]\( q = \frac{1}{2} \)[/tex], we can follow these steps:
1. Understand the relationship:
In a binomial distribution, the variance ([tex]\(\sigma^2\)[/tex]) is given by the formula:
[tex]\[
\sigma^2 = n \cdot p \cdot q
\][/tex]
where:
- [tex]\( n \)[/tex] is the number of trials,
- [tex]\( p \)[/tex] is the probability of success,
- [tex]\( q \)[/tex] is the probability of failure ([tex]\( q = 1 - p \)[/tex]).
2. Identify the given values:
- Variance ([tex]\(\sigma^2\)[/tex]) = 1,
- [tex]\( q = \frac{1}{2} \)[/tex].
3. Determine [tex]\( p \)[/tex]:
Since [tex]\( q = 1 - p \)[/tex],
[tex]\[
p = 1 - q = 1 - \frac{1}{2} = \frac{1}{2}
\][/tex]
4. Substitute the known values into the variance formula:
[tex]\[
1 = n \cdot \frac{1}{2} \cdot \frac{1}{2}
\][/tex]
Simplify the equation:
[tex]\[
1 = n \cdot \frac{1}{4}
\][/tex]
5. Solve for [tex]\( n \)[/tex]:
[tex]\[
n = 1 \cdot 4 = 4
\][/tex]
Thus, the value of [tex]\( n \)[/tex] is [tex]\( 4 \)[/tex].
So, the correct answer is:
D. 4
