Respuesta :
We are given the following information
Number of people that are married = 17
Number of people that are not married = 17
Total number of people = 17+17 = 34
Number of jurors to be selected = 12
We are asked to find the following probabilities.
(c) 8 married and 4 that are not married.
The total number of ways to select 12 jurors from a total of 34 people is given by
[tex]^nC_r=\frac{n!}{r!(n-r)!}[/tex]We have n = 34 and r = 12
[tex]^{34}C_{12}=\frac{34!}{12!(34-12)!}=548,354,040[/tex]So, there are 548,354,040 ways to select 12 jurors from a total of 34 people.
The number of ways to select 8 married jurors from a total of 17 married people is given by
n = 17 and r = 8
[tex]^{17}C_8=\frac{17!}{8!(17-8)!}=24,310[/tex]So, there are 24,310 ways to select 8 married jurors from a total of 17 married people.
The number of ways to select 4 not married jurors from a total of 17 not married people is given by
n = 17 and r = 4
[tex]^{17}C_4=\frac{17!}{4!(17-4)!}=2,380[/tex]So, there are 2,380 ways to select 4 not married jurors from a total of 17 not married people.
Finally, the probability is
[tex]P(8\; married\; and\; 4\; not\; married)=\frac{^{17}C_8\times^{17}C_4}{^{34}C_{12}}=\frac{24,310\times2,380}{548,354,040}=\frac{57,857,800}{548,354,040}=0.1055[/tex]There is a 0.1055 probability that the jury consists of 8 married and 4 not married people.
(d) 6 married and 6 that are not married.
We have already found the total number of ways to select 12 jurors from a total of 34 people that is 548,354,040
The number of ways to select 6 married jurors from a total of 17 married people is given by
n = 17 and r = 6
[tex]^{17}C_6=\frac{17!}{6!(17-6)!}=12,376[/tex]So, there are 12,376 ways to select 6 married jurors from a total of 17 married people.
The number of ways to select 6 not married jurors from a total of 17 not married people will also be the same that is 12,376
Finally, the probability is
[tex]P(6\; married\; and\; 6\; not\; married)=\frac{^{17}C_6\times^{17}C_6}{^{34}C_{12}}=\frac{12,376\times12,376}{548,354,040}=\frac{153,165,376}{548,354,040}=0.2793[/tex]There is a 0.2793 probability that the jury consists of 6 married and 6 not married people.
