To find the probability that the first marble Lucia picks is red and the second marble is white, we need to follow these steps:
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Calculate the total number of marbles: \[ 4 \text{ (red)} + 3 \text{ (yellow)} + 3 \text{ (blue)} + 2 \text{ (white)} = 12 \text{ marbles} \]
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Find the probability of picking a red marble first: The number of red marbles is 4 out of a total of 12 marbles. Thus, the probability of picking a red marble first is: \[ P(\text{Red first}) = \frac{4}{12} = \frac{1}{3} \]
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Calculate the probability of picking a white marble second after a red marble has been picked: After picking one red marble, there are now:
- Total marbles: \( 12 - 1 = 11 \)
- Red marbles remaining: \( 4 - 1 = 3 \)
- Yellow marbles remaining: 3
- Blue marbles remaining: 3
- White marbles remaining: 2
The probability of picking a white marble second is then the number of white marbles divided by the new total: \[ P(\text{White second | Red first}) = \frac{2}{11} \]
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Combine the probabilities: The probability of both events happening (picking a red marble first and a white marble second) is the product of the two probabilities calculated: \[ P(\text{Red first and White second}) = P(\text{Red first}) \times P(\text{White second | Red first}) = \frac{1}{3} \times \frac{2}{11} \] \[ = \frac{2}{33} \]
Thus, the probability that the first marble is red and the second one is white is: \[ \boxed{\frac{2}{33}} \]