To solve the problem of finding the sample space for the event of picking one even-numbered ball from a lottery where balls are numbered from 1 through 37, we need to identify all the even numbers within this range.
Let's start by understanding what even numbers are. Even numbers are integers that are divisible by 2 without leaving a remainder. In other words, an even number [tex]\( n \)[/tex] satisfies [tex]\( n \mod 2 = 0 \)[/tex].
Given the lottery numbers range from 1 to 37, we now seek all even numbers within this range. To identify these:
1. The smallest even number in this range is 2.
2. The next even number is [tex]\( 2 + 2 = 4 \)[/tex].
3. Continuing this pattern of adding 2, we generate the sequence:
[tex]\[ 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36 \][/tex]
This list consists of all even numbers from 1 to 37.
Therefore, the sample space for picking one even-numbered ball from the set of balls numbered 1 through 37 is:
[tex]\[ \{2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36\} \][/tex]
Comparing this sample space to the given options:
A. [tex]\(\{1, 2, 3, \ldots, 37\}\)[/tex] - This set includes all numbers from 1 to 37, not just the even ones. Hence, it is incorrect.
B. [tex]\(\{18\}\)[/tex] - This set includes only the number 18, which is one of the even numbers, but it does not represent the entire sample space of even numbers within the specified range. Therefore, it is incorrect.
C. [tex]\(\{2, 4, 6, \ldots, 36\}\)[/tex] - This set correctly includes all even numbers from 2 to 36. Therefore, it is the correct answer.
D. [tex]\(\{2\}\)[/tex] - This set includes only the number 2, which is an even number, but it does not represent the entire sample space of even numbers within the specified range. Therefore, it is incorrect.
Thus, the correct answer is:
[tex]\[ \boxed{\{2, 4, 6, \ldots, 36\}} \][/tex]
