Let's solve the problem step-by-step.
1. Identify the different choices available:
- Sandwich choices: 2 options (turkey or ham)
- Fruit choices: 2 options (apple or orange)
- Drink choices: 2 options (bottled water or juice)
2. Calculate the total number of possible combinations:
- The total number of boxed lunch combinations is found by multiplying the number of choices for each item.
[tex]\[
\text{Total combinations} = 2 \times 2 \times 2 = 8
\][/tex]
3. Determine the number of combinations that include an orange:
- Since the fruit can be either an apple or an orange, and there are 4 combinations for each type of fruit, we need to count the combinations specifically for orange.
[tex]\[
\text{Orange combinations} = 2 \text{ (sandwich choices)} \times 2 \text{ (drink choices)} = 4
\][/tex]
4. Find the number of combinations that include an orange and do not include juice (i.e., with bottled water):
- For the fruit to be an orange and the drink to be bottled water, there are 2 options for the sandwich.
[tex]\[
\text{Orange and no juice combinations} = 2 \text{ (sandwich choices)} \times 1 \text{ (bottled water)} = 2
\][/tex]
5. Calculate the probability:
- The probability is the number of favorable outcomes (combinations with orange and bottled water) divided by the total number of possible outcomes (total combinations).
[tex]\[
\text{Probability} = \frac{\text{Orange and no juice combinations}}{\text{Total combinations}} = \frac{2}{8} = \frac{1}{4}
\][/tex]
Hence, the probability that you will get an orange and not get juice in your box is [tex]\(\frac{1}{4}\)[/tex].
So, the correct answer is:
[tex]\(\boxed{\frac{1}{4}}\)[/tex]
