To find the probability that Lana randomly chooses a blue marble, replaces it, and then chooses a purple marble, we can follow these steps:
- Calculate the total number of marbles in the bag, which is given as 15.
- Find the probability of choosing a blue marble on the first draw.
- Since she replaces the marble, the total number of marbles remains the same for the second draw.
- Find the probability of choosing a purple marble on the second draw.
- Multiply the probabilities of the two independent events.
Step 1: Total number of marbles
Total marbles = 15
Step 2: Probability of choosing a blue marble
There are 5 blue marbles.
Probability of blue marble \( P(\text{Blue}) = \frac{5}{15} = \frac{1}{3} \)
Step 3: Probability of choosing a purple marble (after replacing the blue)
There are 2 purple marbles.
Probability of purple marble \( P(\text{Purple}) = \frac{2}{15} \)
Step 4: Overall probability of the two events
Since the events are independent (due to replacement):
\[
P(\text{Blue and Purple}) = P(\text{Blue}) \times P(\text{Purple}) = \left(\frac{1}{3}\right) \times \left(\frac{2}{15}\right)
\]
Calculating this gives:
\[
P(\text{Blue and Purple}) = \frac{1}{3} \times \frac{2}{15} = \frac{2}{45}
\]
So, the probability that she randomly chooses a blue marble, replaces it, and then chooses a purple marble is:
\(\frac{2}{45}\)