To find the probability of randomly picking a green marble from the bag, we first need to determine the total number of marbles in the bag.
The contents of the bag are:
- 5 blue marbles
- 3 green marbles
- 2 red marbles
Now, we can calculate the total number of marbles:
\[ \text{Total number of marbles} = 5 + 3 + 2 = 10 \]
Next, we find the number of favorable outcomes, which is the number of green marbles:
\[ \text{Number of green marbles} = 3 \]
The probability of picking a green marble is the number of green marbles divided by the total number of marbles:
\[ P(\text{green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{3}{10} \]
Therefore, the probability of randomly picking a green marble is:
\[ \boxed{\frac{3}{10}} \]