To find the probability that Ryan pulls out a red pen and a black pen without replacement, we can follow these steps:
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Calculate the probability of pulling out a red pen first:
- There are 3 red pens out of 10 total pens.
- Probability of pulling a red pen first = \( \frac{3}{10} \).
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After pulling out a red pen, calculate the probability of pulling out a black pen next:
- After pulling out 1 red pen, there would be 9 pens left (2 red, 4 black, 3 blue).
- Probability of pulling a black pen after pulling a red pen = \( \frac{4}{9} \).
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Now calculate the combined probability:
- Combined probability = Probability of red first * Probability of black second \[ P(\text{Red first, Black second}) = \frac{3}{10} \times \frac{4}{9} = \frac{12}{90} = \frac{2}{15} \]
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Now, calculate the probability of pulling a black pen first and then a red pen:
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Probability of pulling a black pen first = \( \frac{4}{10} \).
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After pulling out 1 black pen, there would again be 9 pens left (3 red, 3 black, 3 blue).
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Probability of pulling a red pen next = \( \frac{3}{9} = \frac{1}{3} \).
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Combined probability for this scenario: \[ P(\text{Black first, Red second}) = \frac{4}{10} \times \frac{1}{3} = \frac{4}{30} = \frac{2}{15} \]
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Add the probabilities of both scenarios: \[ P(\text{Red and Black}) = P(\text{Red first, Black second}) + P(\text{Black first, Red second}) = \frac{2}{15} + \frac{2}{15} = \frac{4}{15} \]
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Convert the probability to a percentage: \[ \text{Percentage} = \left( \frac{4}{15} \times 100 \right) \text{ percent} = \frac{400}{15} \approx 26.67% \]
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Round to the nearest whole number: \[ \approx 27% \]
Therefore, the chances that Ryan pulls out a red pen and a black pen are approximately 27 percent.