The problem asks for the probability of first pulling out a bottle of orange juice and then a bottle of grape juice from an ice chest, without replacing the first bottle. The key information is as follows:

• Orange Juice Bottles: 2
• Grape Juice Bottles: 5
• Apple Juice Bottles: 1
• Total Bottles: $2 + 5 + 1 = 8$

Step 1: Probability of pulling an orange juice bottle first

There are 2 orange juice bottles out of 8 total bottles.

$P(\text{orange juice}) = \frac{2}{8} = \frac{1}{4}$

Step 2: Probability of pulling a grape juice bottle next

After pulling out one orange juice bottle, there are 7 bottles left, 5 of which are grape juice bottles.

$P(\text{grape juice | orange juice}) = \frac{5}{7}$

Step 3: Combined probability

Since the events are sequential and independent (without replacement), the total probability is the product of the two individual probabilities.

$P(\text{orange juice and then grape juice}) = \frac{1}{4} \times \frac{5}{7} = \frac{5}{28}$

Thus, the probability of pulling out a bottle of orange juice followed by a bottle of grape juice is $\frac{5}{28}$.

Do you want more details or have any questions?

Here are 5 related questions to expand on this:

1. What is the probability of pulling two bottles of grape juice in a row?
2. What is the probability of pulling an apple juice bottle first and a grape juice bottle second?
3. What if the order didn’t matter—what would be the probability of pulling one orange and one grape juice bottle?
4. How would the probability change if the first bottle was replaced?
5. How would the probabilities change if the chest had more bottles of orange juice?

Tip: When dealing with probabilities of multiple events without replacement, always reduce the total number of items and adjust the probabilities accordingly.