To find the probability of randomly choosing a gel pen and then a fountain pen (with replacement), we can follow these steps:
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Calculate the probability of choosing a gel pen:
- Total pens: 20 (4 fountain pens + 10 gel pens + 6 ballpoint pens)
- Probability of choosing a gel pen = Number of gel pens / Total number of pens = 10 / 20 = 1/2
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Calculate the probability of choosing a fountain pen:
- Since we are replacing the pen, the total number of pens remains the same.
- Probability of choosing a fountain pen = Number of fountain pens / Total number of pens = 4 / 20 = 1/5
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Calculate the overall probability of both events occurring:
- Since these two events are independent (the first pen is replaced, keeping the total count constant), we multiply their probabilities: \[ P(\text{gel pen and then fountain pen}) = P(\text{gel pen}) \times P(\text{fountain pen}) = \left(\frac{1}{2}\right) \times \left(\frac{1}{5}\right) = \frac{1}{10} \]
Thus, the probability of randomly choosing a gel pen, replacing it, and then choosing a fountain pen is \(\frac{1}{10}\).
The correct response is \( \frac{1}{10} \).