Respuesta :
Answer:
The probability of the event that each of the 18 customers are able to purchases their desired drink is = 0.4653.
Step-by-step explanation:
Given that,
Only regular Coke and diet Coke is dispensed a soft-drink machine .
Diet drink that purchases from this machine is 70% of all purchases.
There are total 18 customers, and X represents number of customer who choose diet coke and Y represents number of customer who choose regular coke.
There are only 12 cans of each types available and X+Y= 18.
The combination of (X,Y) are
(12,6),(11,7),(10,8),(9,9),(8,10),(7,11),(6,12)
The number of diet coke varies 6 to 12
The random variable X follows the binomial distribution of n=18 and p=70%=0.70
[tex]b(x;n,p)=\left(\begin{array}{c}n\\x\end{array}\right) p^x(1-p)^{n-x}[/tex]
The probability that each customers get theirs favorable drink is
=P(6≤X≤12)
=P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)+P(X=12)
[tex]P(X=6)=\left(\begin{array}{c}18\\6\end{array}\right) (0.70)^6(1-0.70)^{18-6}[/tex]
=0.00116
[tex]P(X=7)=\left(\begin{array}{c}18\\7\end{array}\right) (0.70)^7(1-0.70)^{18-7}[/tex]
=0.00464
[tex]P(X=8)=\left(\begin{array}{c}18\\8\end{array}\right) (0.70)^8(1-0.70)^{18-8}[/tex]
=0.0149
[tex]P(X=9)=\left(\begin{array}{c}18\\9\end{array}\right) (0.70)^9(1-0.70)^{18-9}[/tex]
=0.0386
[tex]P(X=10)=\left(\begin{array}{c}18\\10\end{array}\right) (0.70)^{10}(1-0.70)^{18-10}[/tex]
=0.0811
[tex]P(X=11)=\left(\begin{array}{c}18\\11\end{array}\right) (0.70)^{11}(1-0.70)^{18-11}[/tex]
=0.1376
[tex]P(X=12)=\left(\begin{array}{c}18\\12\end{array}\right) (0.70)^{12}(1-0.70)^{18-12}[/tex]
=0.1873
∴P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)+P(X=11)+P(X=12)
=0.4653
The probability that each of the 18 is able to purchases the type of drink desired is = 0.4653.
