Question

Let a random variable X have a binomial distribution with mean 8 and standard deviation 2. If $P(X<2) = \frac{k}{216}, \text{ then the value of } k \text{ is } \_\_\_$.

• A. 17
• B. 16
• C. 136
• D. 137

Hint:

For binomial distribution problems, relate the mean and standard deviation using $np$ and $\sqrt{np(1-p)}$ to find unknowns.

Answer 1

The Correct Option is A

We know that the mean and standard deviation for a binomial distribution are given by: \[ \mu = np \quad \text{and} \quad \sigma = \sqrt{np(1-p)} \] From the problem, $\mu = 8$ and $\sigma = 2$. We have the equations: \[ 8 = np \quad \text{(1)} \] \[ 2 = \sqrt{np(1-p)} \quad \text{(2)} \] Using equation (1) to express $n$ in terms of $p$, we get: \[ n = \frac{8}{p} \] Substitute this into equation (2), solve for $p$, and then calculate the value of $k$. After calculation, we find $k = 17$.