Step 1 :First, we calculate the total number of doughnuts in the box. The box contains 3 chocolate glazed, 5 Boston crème, and 7 powdered sugar doughnuts. So, the total number of doughnuts is \(3 + 5 + 7 = 15\).
Step 2 :Next, we calculate the probability of Carlos selecting a Boston crème doughnut first. The number of Boston crème doughnuts is 5. So, the probability is \(\frac{5}{15} = 0.333\).
Step 3 :After Carlos selects a Boston crème doughnut, the total number of doughnuts in the box decreases by one, becoming 14. We then calculate the probability of him selecting a chocolate glazed doughnut. The number of chocolate glazed doughnuts is 3. So, the probability is \(\frac{3}{14} = 0.214\).
Step 4 :The final probability is the product of the two probabilities calculated above. So, the final probability is \(0.333 \times 0.214 = 0.071\).
Step 5 :Final Answer: The probability that Carlos randomly selects a Boston crème first and then a chocolate glazed doughnut is \(\boxed{0.071}\).