To determine the probability that Jill grabs two grape juice boxes in succession without replacement, we need to go step-by-step through the problem.
1. Total Number of Juice Boxes:
First, we count the total number of juice boxes in the cooler.
- Apple juice boxes: 3
- Mixed fruit juice boxes: 1
- Grape juice boxes: 2
- Tropical fruit juice boxes: 4
Adding these together:
[tex]\[
3 + 1 + 2 + 4 = 10 \text{ juice boxes}
\][/tex]
2. Number of Grape Juice Boxes:
We see that there are 2 grape juice boxes.
3. Probability of Grabbing the First Grape Juice Box:
Since there are 10 boxes in total and 2 of them are grape, the probability of grabbing a grape juice box first is:
[tex]\[
\frac{2}{10} = 0.2
\][/tex]
4. Probability of Grabbing the Second Grape Juice Box Given the First Was Grape:
If one grape juice box has already been picked, we are left with 1 grape juice box out of a total of 9 remaining boxes (10 total boxes minus 1 picked box).
[tex]\[
\frac{1}{9} \approx 0.1111
\][/tex]
5. Calculating the Combined Probability:
We multiply the probability of the first event by the probability of the second event to get the combined probability of both events happening in sequence:
[tex]\[
\text{Combined Probability} = \left( \frac{2}{10} \right) \times \left( \frac{1}{9} \right) = \frac{2}{90} = \frac{1}{45} \approx 0.0222
\][/tex]
6. Conclusion:
The correct probability that Jill will grab two grape juice boxes without replacement is:
[tex]\[
\boxed{\frac{1}{45}}
\][/tex]
This matches the given multiple-choice fraction [tex]$\frac{1}{45}$[/tex].
