To find the probability of randomly choosing 1 orange juice and then 1 apple juice without replacement, we can follow these steps:
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Count the Total Number of Bottles: We have a total of 6 bottles of juice:
- 1 orange juice
- 2 apple juices
- 2 cranberry juices
- 1 grape juice
Thus, the total number of bottles is \(1 + 2 + 2 + 1 = 6\).
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Calculate the Probability of Choosing Orange Juice First: There is 1 orange juice bottle out of 6 total bottles. Therefore, the probability \(P(\text{Orange first})\) is: \[ P(\text{Orange first}) = \frac{1}{6} \]
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Calculate the Probability of Choosing Apple Juice Second: After selecting the orange juice, we do not replace it, so there are now 5 bottles left. Out of these, there are still 2 apple juices remaining. Thus, the probability \(P(\text{Apple second | Orange first})\) is: \[ P(\text{Apple second | Orange first}) = \frac{2}{5} \]
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Calculate the Combined Probability: The total probability of both events happening (choosing orange juice first and then choosing apple juice) is given by the multiplication of the individual probabilities: \[ P(\text{Orange first and Apple second}) = P(\text{Orange first}) \times P(\text{Apple second | Orange first}) = \frac{1}{6} \times \frac{2}{5} \] \[ P(\text{Orange first and Apple second}) = \frac{1 \times 2}{6 \times 5} = \frac{2}{30} = \frac{1}{15} \]
Therefore, the probability of randomly choosing 1 orange juice and then 1 apple juice without replacing the first juice is \(\frac{1}{15}\).