Answer:
The probability is [tex]P[ D n R] = 0.196[/tex]
Step-by-step explanation:
From the question we are told that
The number of Democrats is [tex]D = 8[/tex]
The number of republicans is [tex]R = 8[/tex]
The number of ways of selecting selecting two Democrats and four Republicans.
[tex]N = \left {D} \atop {}} \right. C_2 * \left {R} \atop {}} \right. C_1[/tex]
Where C represents combination
substituting values
[tex]N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1[/tex]
[tex]N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8!}{(8-2)! 2!} * \frac{8! }{(8-4)! 1 !}[/tex]
=> [tex]N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8!}{(6)! 2!} * \frac{8! }{(6)! 1 !}[/tex]
=> [tex]N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8 * 7 * 6!}{(6)! 2!} * \frac{8*7 *6! }{(6)! 1 !}[/tex]
=> [tex]N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = \frac{8 * 7 }{ 2*1 } * \frac{8*7 }{ 1 *1 }[/tex]
=> [tex]N = 1568[/tex]
The total number of ways of selecting the committee of six people is
[tex]Z = \left {D+R} \atop {}} \right. C_6[/tex]
substituting values
[tex]Z = \left {8+8} \atop {}} \right. C_6[/tex]
[tex]Z= \left {16} \atop {}} \right. C_6[/tex]
substituting values
[tex]Z= \left {16} \atop {}} \right. C_6 = \frac{16! }{(16-6) ! 6!}[/tex]
[tex]Z= \left {16} \atop {}} \right. C_6 = \frac{16 *15 *14 * 13 * 12 * 11 * 10! }{10 ! 6!}[/tex]
[tex]Z= \left {16} \atop {}} \right. C_6 = \frac{16 *15 *14 * 13 * 12 * 11 }{6* 5 * 4 * 3 * 2 * 1}[/tex]
[tex]Z= \left {16} \atop {}} \right. C_6 = 8008[/tex]
The probability of selecting two Democrats and four Republicans is mathematically represented as
[tex]P[ D n R] = \frac{N}{Z}[/tex]
substituting values
[tex]P[ D n R] = \frac{1568}{8008}[/tex]
[tex]P[ D n R] = 0.196[/tex]
