To find the probability of drawing two green marbles from the bag without replacement, we first need to determine the total number of marbles and then the specific number of green marbles.
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Total number of marbles: \[ \text{Total marbles} = 5 \text{ (red)} + 7 \text{ (blue)} + 6 \text{ (green)} = 18 \text{ marbles} \]
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Total number of green marbles: There are 6 green marbles in the bag.
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Calculating the probability: We will calculate the probability of drawing two green marbles in succession without replacement.
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The probability of drawing the first green marble: \[ P(\text{1st green}) = \frac{6}{18} \]
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After drawing one green marble, there will be 5 green marbles left and the total number of marbles will be 17. The probability of then drawing a second green marble is: \[ P(\text{2nd green} | \text{1st green}) = \frac{5}{17} \]
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Finding the combined probability: The probability of both events happening in sequence (drawing two green marbles) can be calculated by multiplying the probabilities of each individual event: \[ P(\text{2 green marbles}) = P(\text{1st green}) \times P(\text{2nd green} | \text{1st green}) = \frac{6}{18} \times \frac{5}{17} \]
Simplifying: \[ P(\text{2 green marbles}) = \frac{1}{3} \times \frac{5}{17} = \frac{5}{51} \]
Thus, the exact probability that both marbles drawn will be green is: \[ \boxed{\frac{5}{51}} \]