Question

A group of 40 students from your school is part of the audience for a TV game show. The total number of people in the audience is 130. What is the theoretical probability of 5 students from your school being selected as contestants out of 10 possible contestant spots? P(5 students selected) ______

Answer 1

Step 1

Ways to choose 5 from 40 multiplied by ways to choose 5 from the other 90 and number of ways to choose 10 in all from 130 as the denominator

Step 2:

Apply combination formula

$^nC_r\text{ = }\frac{n!}{(n\text{ - r)!r!}}$

Step 3:

Ways to choose 5 from 40

$^{40}C_5\text{ = }\frac{40!}{(40-5)!5!}\text{ = 658008 way}$

Step 4

ways to choose 5 from the other 90

$^{90}C_5\text{ = }\frac{90!}{(90-5)!5!}\text{ = 43949268 ways}$

Step 5

ways to choose 10 in all from 130

$^{130}C_{10}\text{ = }\frac{130!}{(130-10)!10!}\text{ = 266401260900000 ways}$

Step 6

Ways to choose 5 from 40 multiplied by ways to choose 5 from the other 90 and number of ways to choose 10 in all from 130 as the denominator

$\begin{gathered} \text{Probability of selecting 5 students from your school} \\ =\text{ }\frac{\text{658008 }\times\text{43949268 }}{\text{266401260900000 }} \\ =\text{ 0.109} \end{gathered}$