Question

The cafeteria creates pre-made boxed lunches with equal numbers of the following items:

a sandwich made with either white or wheat bread and either roast beef or bologna
a snack that is either chips, popcorn, or pretzels
a drink that is either bottled water or juice
If Gretchen randomly chooses one of the boxed lunches, what is the probability that she will get a roast beef sandwich and popcorn in her box?

Answer 1

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

1. **Sandwiches**:
- Bread options: 2 (white or wheat)
- Meat options: 2 (roast beef or bologna)

Total sandwich combinations: \(2 \times 2 = 4\)

2. **Snacks**:
- Snack options: 3 (chips, popcorn, or pretzels)

Total snack combinations: \(3\)

3. **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)

1. **Sandwich**:
- Specifically, roast beef sandwich (1 way).

2. **Snack**:
- Specifically, popcorn (1 way).

3. **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 \(\frac{1}{12}\).