To find the probability of randomly picking a green marble, we first need to determine the total number of marbles in the bag.
The bag contains:
- 5 blue marbles
- 3 green marbles
- 2 red marbles
Calculating the total number of marbles:
\[ 5 + 3 + 2 = 10 \]
Next, we find the number of favorable outcomes, which is the number of green marbles:
\[ 3 \text{ green marbles} \]
Now, we can calculate the probability of picking a green marble by using the formula:
\[ P(\text{green marble}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{3}{10} \]
Since \( \frac{3}{10} \) is already in its simplest form, the probability of randomly picking a green marble is:
\[ \boxed{\frac{3}{10}} \]