Final answer:
The probability that the first name on the list is not selected for a public opinion survey is 4/7, calculated by the number of ways to choose three names out of the remaining six divided by the total ways to choose three out of seven.
Step-by-step explanation:
To calculate the probability that the first name on the list is NOT selected for the survey, we need to consider the different possible scenarios for selecting three names out of seven. When we choose three names out of seven without replacement, the number of ways to do this is given by the combination formula, which is C(n, k) = n! / (k!(n - k)!), where n is the total number of items to choose from, and k is the number of items to choose.
In this case, n is 7 (the total number of names), and k is 3 (the number of names we want to select). So, there are a total of 7! / (3!(7 - 3)!) = 35 possible ways to choose three names.
To find the scenarios where the first name is not included, we can consider the remaining six names and find out how many combinations of three can be made from them, which is 6! / (3!(6 - 3)!) = 20.
Therefore, the probability that the first name is not selected is given by the number of combinations without the first name divided by the total number of combinations, which is 20/35. This simplifies to 4/7, representing the probability that the first name on the list is not chosen for the survey.