Lana has a bag of 15 marbles. There are 2 black marbles, 5 blue, 3 red, 3 green, and 2 purple. What is the probability that she randomly chooses a blue marble, replaces it, and then chooses a purple marble? Write your answer as a fraction in the simplest form.(1 point)

Question

The probability is
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Answer 1

To calculate the probability that Lana randomly chooses a blue marble, replaces it, and then chooses a purple marble, follow these steps:

1. Determine the total number of marbles.
   Lana has a total of:

   • 2 black
   • 5 blue
   • 3 red
   • 3 green
   • 2 purple
     So, the total number of marbles = \( 2 + 5 + 3 + 3 + 2 = 15 \).

2. Calculate the probability of choosing a blue marble.
   The probability of choosing a blue marble: \[ P(\text{blue}) = \frac{\text{Number of blue marbles}}{\text{Total number of marbles}} = \frac{5}{15} = \frac{1}{3}. \]

3. Calculate the probability of choosing a purple marble.
   The probability of choosing a purple marble: \[ P(\text{purple}) = \frac{\text{Number of purple marbles}}{\text{Total number of marbles}} = \frac{2}{15}. \]

4. Combine the probabilities.
   Since Lana replaces the blue marble before choosing the next one, the two events are independent. Therefore, the combined probability is: \[ P(\text{blue and then purple}) = P(\text{blue}) \times P(\text{purple}) = \frac{1}{3} \times \frac{2}{15} = \frac{2}{45}. \]

Thus, the probability that she randomly chooses a blue marble, replaces it, and then chooses a purple marble is \( \frac{2}{45} \).

The final answer is: \[ \frac{2}{45}. \]