To find the probability of drawing a green marble and then a black marble from the bag without replacement, we first need to determine the total number of marbles and the number of each color:
- Total marbles = 25
- Purple marbles = 6
- Yellow marbles = 7
- Green marbles = 2
- Black marbles = 25 - (6 + 7 + 2) = 25 - 15 = 10
Now we can calculate the probability:
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Probability of drawing a green marble first: \[ P(\text{Green}) = \frac{\text{Number of green marbles}}{\text{Total number of marbles}} = \frac{2}{25} \]
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After drawing one green marble, the total number of marbles decreases to 24, and the number of black marbles remains 10.
Probability of drawing a black marble second: \[ P(\text{Black | Green drawn}) = \frac{\text{Number of black marbles}}{\text{Total number of marbles remaining}} = \frac{10}{24} = \frac{5}{12} \]
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Now, we can find the combined probability of both events happening: \[ P(\text{Green then Black}) = P(\text{Green}) \times P(\text{Black | Green drawn}) = \frac{2}{25} \times \frac{5}{12} \]
\[ P(\text{Green then Black}) = \frac{2 \times 5}{25 \times 12} = \frac{10}{300} = \frac{1}{30} \]
Thus, the probability of drawing a green marble first and a black marble second without replacement is \(\frac{1}{30}\).
If you're provided with options A, B, C, and D, you can choose the one that corresponds to \(\frac{1}{30}\).