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There were 3 apple, 1 mixed fruit, 2 grape, and 4 tropical fruit juice boxes in the cooler at the picnic. What is the probability that, when Jill reaches into the cooler to grab two juice boxes without replacing them, she grabs two that are grape?

A. [tex]$\frac{1}{100}$[/tex]
B. [tex]$\frac{1}{50}$[/tex]
C. [tex]$\frac{1}{45}$[/tex]
D. [tex]$\frac{1}{25}$[/tex]



Answer :

GinnyAnswer
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].

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