Answer:
a) 0.1317
b) 0.7413
c) 0.2731
d) 0.1561
Step-by-step explanation:
Let's start defining the random variable.
X : ''The number of people that she interviews''
X ~ Bi (n,p)
This means, X can be modeled as a Binomial random variable where p is the success probability and n is the number of independent Bernoulli experiments that take place. In this case, n is the number of interviews that she makes.
The probability function for X is :
[tex]P(X=x)=(nCx).p^{x}(1-p)^{n-x}[/tex]
Where nCx is the combinatorial number defines as :
[tex]nCx=\frac{n!}{x!(n-x)!}[/tex]
For a)
We are looking for the probability of she interviews 5 people over 5 she interviews ⇒
[tex]P(X=5)=(5C5)(\frac{2}{3})^{5}(\frac{1}{3})^{0}=\frac{32}{243}=0.1317[/tex]
b) In this part , we are looking for P(X ≥ 5) when n = 8 (Number of people she interviews)
[tex]P(X\geq 5)= P(X=5)+P(X=6)+P(X=7)+P(X=8)[/tex]
[tex]P(X\geq 5)=(8C5)(\frac{2}{3})^{5}(\frac{1}{3})^{3}+(8C6)(\frac{2}{3})^{6}(\frac{1}{3})^{2}+(8C7)(\frac{2}{3})^{7}(\frac{1}{3})^{1}+(8C8)(\frac{2}{3})^{8}(\frac{1}{3})^{0}=0.7413[/tex]
c) Interview 6 people of 8 :
[tex]P(X=6)=(8C6)(\frac{2}{3})^{6}(\frac{1}{3})^{2}=0.2731[/tex]
d) Interview 7 people of 8 :
[tex]P(X=7)=(8C7)(\frac{2}{3})^{7}(\frac{1}{3})^{1}=0.1561[/tex]
