Let's analyze Nick's suggestion step by step.
1. Total Numbers to Assign: There are 20 numbers available, ranging from 1 to 20.
2. Number of Friends: There are 3 friends - Nick, Sarah, and Mike.
3. Numbers Per Friend: To check how many numbers each friend will get, we divide the total numbers by the number of friends:
[tex]\[
\text{Numbers assigned to each friend} = \frac{20}{3} \approx 6.67
\][/tex]
This resulting number indicates that each friend would theoretically get around 6.67 numbers.
4. Fairness of Assignment:
- For the assignment of numbers to be fair, each friend must receive an equal number of integers.
- Since the division of 20 by 3 results in a non-integer (6.67), we cannot divide the 20 numbers into three equal integer parts.
Thus, the distribution of numbers would look like this:
[tex]\[
\begin{tabular}{|c|c|}
\hline
Friend & Assigned Numbers \\
\hline
Nick & 6 or 7 numbers \\
\hline
Sarah & 6 or 7 numbers \\
\hline
Mike & 6 or 7 numbers \\
\hline
\end{tabular}
\][/tex]
One or more friends will inevitably end up with 7 numbers while the others will have 6. As it's impossible to assign exactly 6.67 numbers per friend, this discrepancy makes the process unfair.
When Natalie selects a number from 1 to 20, the friend with 7 numbers is more likely to be chosen than the friends with 6 numbers. Hence, Nick's suggestion is not fair because the allocation of numbers is not equal.
The true result:
[tex]\[
(6.666666666666667, \text{False})
\][/tex]
indicates that the exact division is approximately 6.67 per friend and that achieving a fair division is not possible. Therefore, Nick's suggestion to use a random number generator to assign numbers is unfair.
