To determine the probability of randomly picking an orange hat, we first need to assess the overall possibilities available when choosing a piece of clothing and a color.
- Choices of clothing: There are 4 pieces of clothing: shirt, pants, socks, and hat.
- Choices of color: There are 3 colors: purple, blue, and orange.
Now, we can calculate the total number of possible combinations of clothing and color:
\[ \text{Total combinations} = (\text{Number of clothing pieces}) \times (\text{Number of colors}) = 4 \times 3 = 12 \]
Next, to find the probability of picking an orange hat, we need to identify how many favorable outcomes exist for this specific event:
- There is only 1 favorable outcome where you pick an orange hat (hat + orange).
Now, we can compute the probability:
\[ \text{Probability of picking an orange hat} = \frac{\text{Number of favorable outcomes}}{\text{Total combinations}} = \frac{1}{12} \]
Thus, the probability that you'll end up with an orange hat is
\[ \boxed{\frac{1}{12}}. \]