To find the probability that all three marbles drawn from the bag are red, we start by determining the total number of marbles in the bag and then the number of favorable outcomes (drawing three red marbles).
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Total Number of Marbles:
- Red marbles: 7
- Blue marbles: 5
- Green marbles: 3
- Total = \( 7 + 5 + 3 = 15 \) marbles.
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Number of Ways to Draw 3 Red Marbles: The number of ways to choose 3 red marbles from the 7 available can be computed using the combination formula: \[ \binom{n}{k} = \frac{n!}{k!(n-k)!} \] For our case, we want to choose 3 out of 7: \[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = \frac{210}{6} = 35. \]
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Total Number of Ways to Draw 3 Marbles from 15: We also calculate the total number of ways to select 3 marbles from the 15 marbles: \[ \binom{15}{3} = \frac{15!}{3!(15-3)!} = \frac{15 \times 14 \times 13}{3 \times 2 \times 1} = \frac{2730}{6} = 455. \]
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Calculating the Probability: The probability that all three marbles drawn are red is given by the ratio of the number of favorable outcomes (choosing 3 red marbles) to the total outcomes (choosing 3 marbles from 15): \[ P(\text{all red}) = \frac{\text{Number of ways to choose 3 red}}{\text{Total number of ways to choose 3 marbles}} = \frac{\binom{7}{3}}{\binom{15}{3}} = \frac{35}{455}. \]
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Simplifying the Probability: To simplify \(\frac{35}{455}\):
- We find the greatest common divisor (GCD) of 35 and 455, which is 35.
- Dividing both the numerator and the denominator by 35 gives: \[ \frac{35 \div 35}{455 \div 35} = \frac{1}{13}. \]
Thus, the exact probability that all three marbles drawn will be red is \(\boxed{\frac{1}{13}}\).