A cooler is filled with different juices. There are 6 bottles of juice: 1 orange juice, 2 apple juice, 2 cranberry juice, and 1 grape juice. What is the probability of randomly choosing 1 orange juice and then 1 apple juice without replacing the first juice?

Answer 1

To find the probability of randomly choosing 1 orange juice and then 1 apple juice without replacement, we can follow these steps:

1. 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 \(1 + 2 + 2 + 1 = 6\).

2. 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: \[ P(\text{Orange first}) = \frac{1}{6} \]

3. 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: \[ P(\text{Apple second | Orange first}) = \frac{2}{5} \]

4. 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 \(\frac{1}{15}\).