Question

In a binomial distribution, the probability of getting a success is \(\frac{1}{3}\) and the standard deviation is 4. Then its mean is :

• A. 8
• B. 24
• C. 16
• D. 32

Answer 1

The Correct Option is B

To find the mean of a binomial distribution, we use the formula for the mean \(\mu\) given by:

\(\mu=n \cdot p\)

where \(n\) is the number of trials and \(p\) is the probability of success. The standard deviation \(\sigma\) of a binomial distribution is given by:

\(\sigma=\sqrt{n \cdot p \cdot (1-p)}\)

We know that \(p=\frac{1}{3}\) and \(\sigma=4\). Substituting these values into the standard deviation formula, we can solve for \(n\):

\(4=\sqrt{n \cdot \frac{1}{3} \cdot \left(1-\frac{1}{3}\right)}\)

\(4=\sqrt{n \cdot \frac{1}{3} \cdot \frac{2}{3}}\)

\(4=\sqrt{\frac{2n}{9}}\)

\(16=\frac{2n}{9}\)

\(2n=144\)

\(n=72\)

Now we substitute \(n=72\) and \(p=\frac{1}{3}\) into the mean formula:

\(\mu=72 \cdot \frac{1}{3}\)

\(\mu=24\)

Thus, the mean of the binomial distribution is 24.