Question

Instructor is to examine a population of 20 students to check for homework. If 15 have done their homework and if sample of 4 students is to be randomly chosen, what is the probability that exactly 3 will have done their homework? What is the probability that exactly 3 will have done their homework

Answer 1

Let x be the number of students who have done their homework

Given Population(N)= 20 ; Sample(n)=4 ; done homework (m)= m

Then the proof of x is

$P(x=k)= \frac{\begin{pmatrix} m \\ k \end{pmatrix} \begin{pmatrix} N-m \\ n-k \end{pmatrix}}{\begin{pmatrix} N \\ n \end{pmatrix}}= \frac{\begin{pmatrix} 15 \\ k \end{pmatrix} \begin{pmatrix} 20-m \\ 4-k \end{pmatrix}}{\begin{pmatrix} 20 \\ 4 \end{pmatrix}}$

P( exactly 3 will have done their work)

$= P(x=3)\\ = \frac{\begin{pmatrix} 15 \\ 3 \end{pmatrix} \begin{pmatrix} 20-15 \\ 4-3 \end{pmatrix}}{\begin{pmatrix} 20 \\ 4 \end{pmatrix}}\\ = \frac{(\frac{15!}{(15-3)! 3!}) (\frac{5!}{(5-1)! 1!})}{(\frac{20!}{(20-4)! 4!})}\\ =\frac{455*5}{4845}\\ =0.47$