Determine the indicated probability for a binomial experiment with the given number of trials n=12 and the given success probability p=0.8. Then find the mean, variance, and standard deviation.

a) P(9)=

b)${\mu }_X$ =

c) ${\sigma }_X^2$ =

d) ${\sigma }_X$ =

Answer :

Given :

Probability of success $(p)=0.8$

Sample size $(n)=12$

Solution :

a) The probability that x exactly 9 :

$$\text{P}(x=9)=(\genfrac{}{}{0ex}{}{n}{x})\cdot p^x\cdot (1-p{)}^{n-x}$$ $$=(\genfrac{}{}{0ex}{}{12}{9})\cdot {0.8}^9\cdot (1–0.8{)}^{12-9}$$ $$=0.2362$$

b) The mean : ${\mu }_x=n\cdot p=12\cdot 0.8=9.6$

c) The variance : ${\sigma }_x^2=n\cdot p(1-p)=12\cdot 0.8(1-0.8)=1.92$

d) The standard deviation : ${\sigma }_x=\sqrt{n\cdot p(1-p)}=\sqrt{12\cdot 0.8(1-0.8)}=1.386$