To find the probability of selecting a red marble first and then a blue marble without replacing the red marble, we can follow these steps:
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Calculate the total number of marbles: There are 10 marbles in total (2 red + 3 yellow + 5 blue).
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Probability of selecting a red marble: The probability of selecting one of the 2 red marbles is: \[ P(\text{Red}) = \frac{2 \text{ (red marbles)}}{10 \text{ (total marbles)}} = \frac{2}{10} = \frac{1}{5} \]
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After selecting a red marble, calculate the new total of marbles: After one red marble has been removed, there are now 9 marbles left in total (1 red + 3 yellow + 5 blue).
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Probability of selecting a blue marble after selecting a red marble: The probability of selecting one of the 5 blue marbles is: \[ P(\text{Blue | Red}) = \frac{5 \text{ (blue marbles)}}{9 \text{ (remaining marbles)}} = \frac{5}{9} \]
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Calculate the combined probability: To find the overall probability of both events happening in sequence (selecting a red marble and then a blue marble), we multiply the probabilities of each step: \[ P(\text{Red then Blue}) = P(\text{Red}) \times P(\text{Blue | Red}) = \frac{1}{5} \times \frac{5}{9} = \frac{5}{45} = \frac{1}{9} \]
Therefore, the probability of selecting a red marble and then selecting a blue marble without replacement is \(\frac{1}{9}\).
The correct option from the provided choices is: 191 ninth