At a game show, there are 8 people (including you and your friend) in the front row.
The host randomly chooses 3 people from the front row to be contestants. The order in which they are chosen does not matter.
There are [tex]$B _8 C _3=56$[/tex] total ways to choose the 3 contestants.
What is the probability that you and your friend are both chosen?
A. [tex]$\frac{6}{56}$[/tex]
B. [tex]$\frac{3}{56}$[/tex]
C. [tex]$\frac{2}{56}$[/tex]
Answer (1)
Calculate the number of ways to choose the remaining contestant: ( 1 6 ) = 6 .
The total number of ways to choose 3 contestants from 8 is ( 3 8 ) = 56 .
Calculate the probability: 56 6 .
Simplify the fraction: 56 6 .
Explanation
Understand the problem We are given a problem where a host randomly chooses 3 people out of 8 to be contestants. We want to find the probability that both you and your friend are among the chosen contestants.
Calculate favorable outcomes Since you and your friend must be chosen, we need to choose 1 more person from the remaining 6 people. The number of ways to choose 1 person from 6 is given by the combination formula:
Calculate the combinations ( 1 6 ) = 1 ! ( 6 − 1 )! 6 ! = 1 ! 5 ! 6 ! = 1 × 5 ! 6 × 5 ! = 6
Calculate the probability The total number of ways to choose 3 people from 8 is given as ( 3 8 ) = 56 . The probability that both you and your friend are chosen is the number of ways to choose the remaining contestant divided by the total number of ways to choose 3 contestants:
Final Calculation P ( you and friend chosen ) = ( 3 8 ) ( 1 6 ) = 56 6 = 28 3
State the final answer Therefore, the probability that both you and your friend are chosen is 56 6 , which simplifies to 28 3 .
Examples
This type of probability calculation is useful in scenarios like selecting a team or committee where certain members must be included. For example, if you're forming a project group and need to ensure that both the team lead and a specific expert are part of the group, you can calculate the probability of this happening given a random selection process. This helps in understanding the likelihood of specific compositions in group formations.
