Question

In a binomial distribution the probability of getting a success is $\frac{1}{4}$ and standard deviation is $3$, then its mean is

• A. $6$
• B. $8$
• C. $10$
• D. $12$

Answer 1

The correct option is D

$12$

Explanation for the correct option.

Find the mean of the binomial distribution.

In a binomial distribution, $n$ is the number of trials, $p$ is the probability of success, and $q$ is the probability of failure.

It is given that probability of getting a success is $\frac{1}{4}$, so $p=\frac{1}{4}$.

Now the value of $q$ can be found using $p+q=1$.

$$\begin{gathered} p+q=1 \\ \Rightarrow \frac{1}{4}+q=1 \\ \Rightarrow q=1-\frac{1}{4} \\ \Rightarrow q=\frac{3}{4} \end{gathered}$$

So the probability of getting a failure is $q=\frac{3}{4}$.

Variance is the square of standard deviation. As the standard deviation is $3$. So the variance is $3^2=9$.

Now, variance is given by $npq$. So equate it with $9$ and put the values of $p$ and $q$ to find the value of $n$.

$$\begin{gathered} npq=9 \\ \Rightarrow n\times \frac{1}{4}\times \frac{3}{4}=9\left[p=\frac{1}{4},q=\frac{3}{4}\right] \\ \Rightarrow n=9\times \frac{16}{3} \\ \Rightarrow n=48 \end{gathered}$$

Now the mean of the distribution is given as $np$. So substitute the values and find the mean.

$\begin{array}{rcl}\mathrm{Mean}& =& np\\ & =& 48\times \frac{1}{4}\left[n=48,p=\frac{1}{4}\right]\\ & =& 12\end{array}$

So the mean of the given binomial distribution is $12$.

Hence, the correct option is D.