To find the probability that Gretchen randomly chooses a boxed lunch that contains a roast beef sandwich and popcorn, we first need to determine the total number of possible boxed lunches and then how many of those contain the specific items.
Step 1: Calculate the total number of boxed lunches
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Sandwiches:
- Bread options: 2 (white or wheat)
- Meat options: 2 (roast beef or bologna)
Total sandwich combinations: \(2 \times 2 = 4\)
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Snacks:
- Snack options: 3 (chips, popcorn, or pretzels)
Total snack combinations: \(3\)
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Drinks:
- Drink options: 2 (bottled water or juice)
Total drink combinations: \(2\)
Now, we can calculate the total number of boxed lunch combinations:
\[ \text{Total boxed lunches} = \text{Total sandwiches} \times \text{Total snacks} \times \text{Total drinks} = 4 \times 3 \times 2 = 24 \]
Step 2: Calculate the number of favorable outcomes (roast beef sandwich and popcorn)
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Sandwich:
- Specifically, roast beef sandwich (1 way).
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Snack:
- Specifically, popcorn (1 way).
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Drink:
- Two drink options still apply (bottled water or juice).
Thus, the number of favorable combinations for a boxed lunch with a roast beef sandwich and popcorn is:
\[ \text{Favorable combinations} = 1 \times 1 \times 2 = 2 \]
Step 3: Calculate the probability
Finally, we can determine the probability that Gretchen randomly selects a boxed lunch with a roast beef sandwich and popcorn using the formula:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{2}{24} = \frac{1}{12} \]
Therefore, the probability that Gretchen will get a roast beef sandwich and popcorn in her box is \(\frac{1}{12}\).