Question

a jury pool has 22 people that are married and 22 people that are not married, from which 12 jurors will be selected. assuming that each person is equally likely to be chosen and that the jury is selected at random without replacement, find the probability that the jury consists of the following. (give answer as a fraction or a decimal out to at least 4 places. if your answer is very small use scientific notation out to 4 decimal places for example 3.3421e - 6.)
(a) all married people.
(b) all not married people.
(c) 8 married and 4 that are not married.
(d) 6 married and 6 that are not married.

Explanation:

Step1: Calculate total number of people

The total number of people in the jury - pool is $22 + 22=44$ people. The number of ways to choose 12 jurors out of 44 is given by the combination formula $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n = 44$ and $r = 12$. So, $C(44,12)=\frac{44!}{12!(44 - 12)!}=\frac{44!}{12!32!}$.

Step2: Calculate probability for part (a)

The number of ways to choose 12 married people out of 22 is $C(22,12)=\frac{22!}{12!(22 - 12)!}=\frac{22!}{12!10!}$. The probability $P(a)$ that all 12 jurors are married is $P(a)=\frac{C(22,12)}{C(44,12)}=\frac{\frac{22!}{12!10!}}{\frac{44!}{12!32!}}=\frac{22!×32!}{44!×10!}\approx1.1374\times10^{- 7}$.

Step3: Calculate probability for part (b)

The number of ways to choose 12 non - married people out of 22 is $C(22,12)=\frac{22!}{12!(22 - 12)!}=\frac{22!}{12!10!}$. The probability $P(b)$ that all 12 jurors are non - married is $P(b)=\frac{C(22,12)}{C(44,12)}=\frac{\frac{22!}{12!10!}}{\frac{44!}{12!32!}}=\frac{22!×32!}{44!×10!}\approx1.1374\times10^{-7}$.

Step4: Calculate probability for part (c)

The number of ways to choose 8 married people out of 22 is $C(22,8)=\frac{22!}{8!(22 - 8)!}=\frac{22!}{8!14!}$, and the number of ways to choose 4 non - married people out of 22 is $C(22,4)=\frac{22!}{4!(22 - 4)!}=\frac{22!}{4!18!}$. The number of favorable cases is $C(22,8)\times C(22,4)$. The probability $P(c)$ is $P(c)=\frac{C(22,8)\times C(22,4)}{C(44,12)}=\frac{\frac{22!}{8!14!}\times\frac{22!}{4!18!}}{\frac{44!}{12!32!}}\approx0.1667$.

Step5: Calculate probability for part (d)

The number of ways to choose 6 married people out of 22 is $C(22,6)=\frac{22!}{6!(22 - 6)!}=\frac{22!}{6!16!}$, and the number of ways to choose 6 non - married people out of 22 is $C(22,6)=\frac{22!}{6!(22 - 6)!}=\frac{22!}{6!16!}$. The number of favorable cases is $C(22,6)\times C(22,6)$. The probability $P(d)$ is $P(d)=\frac{C(22,6)\times C(22,6)}{C(44,12)}=\frac{\frac{22!}{6!16!}\times\frac{22!}{6!16!}}{\frac{44!}{12!32!}}\approx0.2954$.

Answer:

(a) $1.1374\times10^{-7}$
(b) $1.1374\times10^{-7}$
(c) $0.1667$
(d) $0.2954$