To determine the probability that both you and your friend are chosen as contestants out of the 6 people, while knowing that the host will choose 3 people in total, let's break down the problem step by step:
1. Total Number of Ways to Choose 3 Out of 6 People:
The total number of ways to choose 3 people from 6 is given by the combination formula [tex]\({}_6C_3\)[/tex]:
[tex]\[
{}_6C_3 = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!} = 20
\][/tex]
2. Number of Favorable Outcomes:
We need to find the number of ways to choose 3 people such that both you and your friend are included among the chosen ones.
- Since you and your friend must be chosen, we are left with choosing 1 more person out of the remaining 4 people.
- The number of ways to choose 1 person out of 4 is given by the combination formula [tex]\({}_4C_1\)[/tex]:
[tex]\[
{}_4C_1 = \frac{4!}{1!(4-1)!} = \frac{4!}{1! \cdot 3!} = 4
\][/tex]
3. Probability Calculation:
The probability is the ratio of the number of favorable outcomes to the total number of outcomes.
[tex]\[
\text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{{}_4C_1}{{}_6C_3} = \frac{4}{20}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\frac{4}{20}}
\][/tex]
