If the mean of a binomial distribution is 5 and its standard deviation is 2, then the probability of ten successes is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \({}^{25}C_{10}\left(\displaystyle\frac{1}{5}\right)^{10}\left(\displaystyle\frac{4}{5}\right)^{15}\)
For a binomial distribution, mean = np and variance = npq.
\(np = 5\)
\(npq = 2^2 = 4\)
Dividing:
\( \displaystyle q = \frac{4}{5},\quad p = 1 - q = \frac{1}{5} \)
\( \displaystyle n = \frac{5}{p} = 25 \)
Then
\( \displaystyle P(X = 10) = {}^{25}C_{10}\left(\frac{1}{5}\right)^{10}\left(\frac{4}{5}\right)^{15} \)
Tip: The power of p goes with the number of successes (10) and the power of q with the failures (25 − 10 = 15). Option (3) swaps them.