Question

A city council consists of eight Democrats and eight Republicans. If a committee of six people is selected, find the probability of selecting two Democrats and four Republicans.

(Type answer a fraction Simplify your answer.)

Answer 1

The probability is
`P[ D n R] = 0.196`

Explanation:

From the question we are told that

The number of Democrats is
`D = 8`

The number of republicans is
`R = 8`

The number of ways of selecting selecting two Democrats and four Republicans.

`N = \left {D} \atop {}} \right. C_2 * \left {R} \atop {}} \right. C_1`

Where C represents combination

substituting values

`N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1`

`N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = (8!)/((8-2)! 2!) * (8! )/((8-4)! 1 !)`

=>
`N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = (8!)/((6)! 2!) * (8! )/((6)! 1 !)`

=>
`N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = (8 * 7 * 6!)/((6)! 2!) * (8*7 *6! )/((6)! 1 !)`

=>
`N = \left {8} \atop {}} \right. C_2 * \left {8} \atop {}} \right. C_1 = (8 * 7 )/( 2*1 ) * (8*7 )/( 1 *1 )`

=>
`N = 1568`

The total number of ways of selecting the committee of six people is

`Z = \left {D+R} \atop {}} \right. C_6`

substituting values

`Z = \left {8+8} \atop {}} \right. C_6`

`Z= \left {16} \atop {}} \right. C_6`

substituting values

`Z= \left {16} \atop {}} \right. C_6 = (16! )/((16-6) ! 6!)`

`Z= \left {16} \atop {}} \right. C_6 = (16 *15 *14 * 13 * 12 * 11 * 10! )/(10 ! 6!)`

`Z= \left {16} \atop {}} \right. C_6 = (16 *15 *14 * 13 * 12 * 11 )/(6* 5 * 4 * 3 * 2 * 1)`

`Z= \left {16} \atop {}} \right. C_6 = 8008`

The probability of selecting two Democrats and four Republicans is mathematically represented as

`P[ D n R] = (N)/(Z)`

substituting values

`P[ D n R] = (1568)/(8008)`

`P[ D n R] = 0.196`