To determine which bag of marbles best models Frosina's experiment, we need to analyze the ratios of the colors of marbles pulled from the bag based on the number of times each color was recorded.
Frosina pulled the marbles a total of:
\[ 18 \text{ (purple)} + 6 \text{ (blue)} + 12 \text{ (yellow)} = 36 \text{ total pulls} \]
Now, let's calculate the proportion for each color based on the total number of pulls:
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Purple: \[ \frac{18}{36} = \frac{1}{2} \]
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Blue: \[ \frac{6}{36} = \frac{1}{6} \]
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Yellow: \[ \frac{12}{36} = \frac{1}{3} \]
Next, we will check the options given in the responses to see which bag has a similar ratio.
Let's analyze each response:
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6 purple, 1 blue, 9 yellow:
- Total = \(6 + 1 + 9 = 16\)
- Ratio: \(\frac{6}{16} : \frac{1}{16} : \frac{9}{16} = \frac{3}{8} : \frac{1}{16} : \frac{9}{16} \), which does not match.
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6 purple, 6 blue, 2 yellow:
- Total = \(6 + 6 + 2 = 14\)
- Ratio: \(\frac{6}{14} : \frac{6}{14} : \frac{2}{14} = \frac{3}{7} : \frac{3}{7} : \frac{1}{7} \), which does not match.
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6 purple, 2 blue, 4 yellow:
- Total = \(6 + 2 + 4 = 12\)
- Ratio: \(\frac{6}{12} : \frac{2}{12} : \frac{4}{12} = \frac{1}{2} : \frac{1}{6} : \frac{1}{3} \), which matches.
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4 purple, 5 blue, 2 yellow:
- Total = \(4 + 5 + 2 = 11\)
- Ratio: \(\frac{4}{11} : \frac{5}{11} : \frac{2}{11}\), which does not match.
The only bag that matches the ratios of \( \frac{1}{2} : \frac{1}{6} : \frac{1}{3} \) derived from Frosina's experiment is therefore the third option:
6 purple, 2 blue, 4 yellow.