The mean of the Binomial distribution \(B\left(4, \displaystyle\frac{1}{3}\right)\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\displaystyle\frac{4}{3}\)
In B(n, p), n = 4 is the number of trials and p = \(\frac{1}{3}\) the probability of success. The mean of a binomial distribution is np:
\( \displaystyle \text{Mean} = np = 4\times\frac{1}{3} = \frac{4}{3} \)
Why the other options are incorrect
Option (2) \(\displaystyle\frac{2}{3}\) — Incorrect. This is q = 1 − p, not the mean.
Option (3) \(\displaystyle\frac{8}{3}\) — Incorrect. This is nq = 4 × \(\frac{2}{3}\), the mean number of failures.
Option (4) \(\displaystyle\frac{1}{3}\) — Incorrect. This is p itself; the mean is n times p.