Question

A group of three undergraduate and five graduate students are available to fill certain student government posts. If four students are to be randomly selected from this group, find the probability that exactly two undergraduates will be among the four chosen.

Answer 1

`Pr = 0.4286`

Explanation:

Given

Let

`U \to\\` Undergraduates

`G \to` Graduates

So, we have:

`U = 3; G =5` -- Total students

`r = 4` --- students to select

Required

`P(U =2)`

From the question, we understand that 2 undergraduates are to be selected; This means that 2 graduates are to be selected.

First, we calculate the total possible selection (using combination)

`^nC_r = (n!)/((n-r)!r!)`

So, we have:

`Total = ^(U + G)C_r`

`Total = ^(3 + 5)C_4`

`Total = ^8C_4`

`Total = (8!)/((8-4)!4!)`

`Total = (8!)/(4!4!)`

Using a calculator, we have:

`Total = 70`

The number of ways of selecting 2 from 3 undergraduates is:

`U = ^3C_2`

`U = (3!)/((3-2)!2!)`

`U = (3!)/(1!2!)`

`U = 3`

The number of ways of selecting 2 from 5 graduates is:

`G = ^5C_2`

`G = (5!)/((5-2)!2!)`

`G = (5!)/(3!2!)`

`G =10`

So, the probability is:

`Pr = (G * U)/(Total)`

`Pr = (10*3)/(70)`

`Pr = (30)/(70)`

`Pr = 0.4286`