To find the probability that Cynara will select a blue pen and then a red pen, we first need to determine the probabilities of selecting each color based on her experiment.
Cynara selected pens in total:
- Red pens: 6
- Blue pens: 10
- Black pens: 14
Total selections = 6 + 10 + 14 = 30.
Now, we can calculate the probabilities for each color:
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Probability of selecting a Blue pen (P(Blue)): \[ P(\text{Blue}) = \frac{\text{Number of Blue pens}}{\text{Total Pens}} = \frac{10}{30} = \frac{1}{3}. \]
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Probability of selecting a Red pen (P(Red)): \[ P(\text{Red}) = \frac{\text{Number of Red pens}}{\text{Total Pens}} = \frac{6}{30} = \frac{1}{5}. \]
Now, since the selections are done with replacement, the two events (selecting a blue pen and then selecting a red pen) are independent. Therefore, we can calculate the joint probability of these two events occurring one after the other by multiplying their probabilities:
\[ P(\text{Blue then Red}) = P(\text{Blue}) \times P(\text{Red}) = \left(\frac{1}{3}\right) \times \left(\frac{1}{5}\right) = \frac{1}{15}. \]
Thus, the probability that Cynara will select a blue pen, return it to the bag, and then select a red pen is:
\[ \boxed{\frac{1}{15}}. \]
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