To determine which list correctly represents the sample space for pulling two slips of paper out of a bucket with replacement, we need to consider all possible outcomes of the two draws. With replacement means that after drawing a slip, it is put back into the bucket before the next draw, so each draw is independent and can result in any of the three colors (Red, Blue, Yellow) regardless of previous draws.
Here's a step-by-step breakdown of our analysis:
1. Identifying Possible Pair Outcomes:
With three different colors (Red, Blue, and Yellow), and since each draw is independent and the slip is replaced, the possible outcomes for each of the two draws are as follows:
- (Red, Red)
- (Red, Blue)
- (Red, Yellow)
- (Blue, Blue)
- (Blue, Red)
- (Blue, Yellow)
- (Yellow, Yellow)
- (Yellow, Red)
- (Yellow, Blue)
2. Comparing Each List:
- List 1:
[tex]\[
\text{Red}, \text{Blue}, \text{Yellow}, \text{Red}, \text{Blue}, \text{Yellow}, \text{Red}, \text{Blue}, \text{Yellow}
\][/tex]
- This list only contains single instances of each color and does not account for pairs.
- List 2:
[tex]\[
\{\text{Red, Red}, \text{Red, Blue}, \text{Red, Yellow}, \text{Blue, Blue}, \text{Blue, Red}, \text{Blue, Yellow}, \text{Yellow, Yellow}, \text{Yellow, Red}, \text{Yellow, Blue}\}
\][/tex]
- This list includes all nine possible pair combinations, accounting for every scenario with replacement.
- List 3:
[tex]\[
\{\text{Red, Blue}, \text{Red, Yellow}, \text{Blue, Red}, \text{Blue, Yellow}, \text{Yellow, Red}, \text{Yellow, Blue}, \text{Red}, \text{Blue}, \text{Yellow}, \text{Red}, \text{Blue}, \text{Yellow}\}
\][/tex]
- This list includes some valid pairs but is missing several combinations (e.g., Yellow, Yellow and Red, Red).
- List 4:
[tex]\[
\{\text{Red, Red, Red}, \text{Red, Blue, Yellow}, \text{Red, Yellow, Blue}, \text{Blue, Red, Yellow}, \text{Blue, Yellow, Red}, \text{Yellow, Red, Blue}, \text{Yellow, Blue, Red}, \text{Yellow, Yellow, Yellow}\}
\][/tex]
- This list contains triplets, which are beyond the required pairs.
3. Conclusion:
List 2 correctly and completely represents the sample space for pulling two slips of paper out of the bucket with replacement. It encompasses all the possible pairs of outcomes that result from drawing two slips, considering each color can repeat due to the replacement process.
Therefore, the correct list is:
[tex]\[
\text{List 2}
\][/tex]
