To find the probability of first choosing a red marble followed by a green marble from the bag, we can break the problem into two parts: the probability of choosing a red marble first, and then the probability of choosing a green marble second.
First, we calculate the total number of marbles in the bag:
- 1 blue
- 2 green
- 2 red
Total marbles = 1 + 2 + 2 = 5 marbles.
Step 1: Probability of choosing a red marble first
The probability of selecting a red marble first is calculated as follows:
\[ P(\text{Red first}) = \frac{\text{Number of red marbles}}{\text{Total number of marbles}} = \frac{2}{5} \]
Step 2: Probability of choosing a green marble second
After removing one red marble (since we are choosing marbles without replacement), we now have:
- 1 blue
- 2 green
- 1 red
Total remaining marbles = 1 + 2 + 1 = 4 marbles.
Now we can calculate the probability of selecting a green marble second:
\[ P(\text{Green second | Red first}) = \frac{\text{Number of green marbles remaining}}{\text{Total remaining marbles}} = \frac{2}{4} = \frac{1}{2} \]
Step 3: Combined Probability
Now, we can find the overall probability of first choosing a red marble and then a green marble by multiplying the probabilities from the two steps:
\[ P(\text{Red first and Green second}) = P(\text{Red first}) \times P(\text{Green second | Red first}) = \frac{2}{5} \times \frac{1}{2} = \frac{2}{10} = \frac{1}{5} \]
Thus, the probability that you will first choose a red marble followed by a green marble is
\[ \boxed{\frac{1}{5}}. \]