To find the probability that both marbles picked are blue, we can use the concept of combinations and the rules of probability.
First, we determine the total number of marbles in the bag:
- Blue marbles: 6
- Red marbles: 4
- Green marbles: 2
Total number of marbles = 6 + 4 + 2 = 12.
Next, we calculate the total number of ways to choose 2 marbles from these 12 marbles. This is given by the combination formula:
\[ \text{C}(n, k) = \frac{n!}{k!(n-k)!} \]
Where \(n\) is the total number of items (marbles), and \(k\) is the number of items to choose. So we calculate:
\[ \text{C}(12, 2) = \frac{12!}{2!(12-2)!} = \frac{12 \times 11}{2 \times 1} = 66. \]
Now, we count the number of ways to choose 2 blue marbles from the 6 blue marbles:
\[ \text{C}(6, 2) = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15. \]
Now we can find the probability of picking 2 blue marbles. The probability \(P\) of an event is given by the ratio of the number of favorable outcomes to the total number of outcomes:
\[ P(\text{both marbles are blue}) = \frac{\text{Number of ways to choose 2 blue marbles}}{\text{Total number of ways to choose 2 marbles}}. \]
Thus, we have:
\[ P(\text{both marbles are blue}) = \frac{15}{66} = \frac{5}{22}. \]
Therefore, the probability that both marbles picked are blue is \(\frac{5}{22}\).