Answered

IDNLearn.com: Your trusted source for accurate and reliable answers. Our platform provides accurate, detailed responses to help you navigate any topic with ease.

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?

A. [tex]$\frac{1}{12}$[/tex]
B. [tex]$\frac{1}{6}$[/tex]
C. [tex]$\frac{1}{2}$[/tex]
D. [tex]$\frac{1}{3}$[/tex]

Sagot :

GinnyAnsweridn
To solve this problem, let's break it down into several logical steps.

### Step 1: Determine Total Possible Combinations

First, we need to calculate the total number of possible combinations that can be created for the boxed lunches.

1. Bread Options:
- White Bread
- Wheat Bread

This gives us [tex]\(2\)[/tex] options.

2. Meat Options:
- Roast Beef
- Bologna

This also gives us [tex]\(2\)[/tex] options.

3. Snack Options:
- Chips
- Popcorn
- Pretzels

This gives us [tex]\(3\)[/tex] options.

4. Drink Options:
- Bottled Water
- Juice

This gives us [tex]\(2\)[/tex] options.

To find the total number of combinations, multiply the number of options for each category together:

[tex]\[ \text{Total Combinations} = 2 \text{ (Bread)} \times 2 \text{ (Meat)} \times 3 \text{ (Snack)} \times 2 \text{ (Drink)} = 24 \][/tex]

### Step 2: Determine Favorable Combinations

Next, we need to calculate the number of favorable combinations that meet the condition of having a roast beef sandwich and popcorn.

1. Condition for Sandwich (Roast Beef):
- White Bread with Roast Beef
- Wheat Bread with Roast Beef

This provides [tex]\(2\)[/tex] favorable options for the sandwich.

2. Condition for Snack (Popcorn):
- Only [tex]\(1\)[/tex] option for popcorn as a snack.

3. Drink Options:
- Bottled Water
- Juice

The drinks can be either, so there are [tex]\(2\)[/tex] options.

Therefore, for roast beef and popcorn, the number of favorable combinations is:

[tex]\[ \text{Favorable Combinations} = 2 \text{ (Sandwich)} \times 1 \text{ (Snack)} \times 2 \text{ (Drink)} = 4 \][/tex]

### Step 3: Calculate the Probability

Finally, we calculate the probability that Gretchen will randomly choose a boxed lunch with a roast beef sandwich and popcorn. Probability is given by:

[tex]\[ \text{Probability} = \frac{\text{Number of Favorable Combinations}}{\text{Total Number of Combinations}} = \frac{4}{24} = \frac{1}{6} \][/tex]

Therefore, the probability that Gretchen will get a boxed lunch with a roast beef sandwich and popcorn is [tex]\(\frac{1}{6}\)[/tex].

So, the correct answer is:

[tex]\[ \boxed{\frac{1}{6}} \][/tex]
Thank you for joining our conversation. Don't hesitate to return anytime to find answers to your questions. Let's continue sharing knowledge and experiences! For trustworthy answers, visit IDNLearn.com. Thank you for your visit, and see you next time for more reliable solutions.