To find the probability that all three marbles drawn from the bag are red, we first need to determine the total number of marbles in the bag and the number of ways to draw 3 red marbles.
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Total number of marbles:
- Red marbles = 6
- Blue marbles = 8
- Green marbles = 4
Total marbles = 6 + 8 + 4 = 18
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Ways to choose 3 red marbles: The number of ways to choose 3 red marbles from 6 is given by the combination formula \( C(n, k) = \frac{n!}{k!(n-k)!} \):
\[ C(6, 3) = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = \frac{120}{6} = 20 \]
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Total ways to choose any 3 marbles from 18:
The number of ways to choose 3 marbles from a total of 18 is:
\[ C(18, 3) = \frac{18!}{3!(18-3)!} = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = \frac{4896}{6} = 816 \]
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Probability that all three drawn marbles are red: The probability \( P \) is given by the ratio of the number of ways to choose 3 red marbles to the total number of ways to choose any 3 marbles:
\[ P = \frac{C(6, 3)}{C(18, 3)} = \frac{20}{816} = \frac{5}{204} \]
Thus, the exact probability that all three marbles drawn will be red is
\[ \boxed{\frac{5}{204}}. \]