Problem 17

To solve this problem, let's break it down step by step, following the given information and the solution approach closely.

Step 1: Identify the total number of classes in the morning and afternoon.
- In the morning, there are 4 classes: mathematics, Chinese, politics, and geography.
- In the afternoon, there are 3 classes: English, history, and physical education.

Step 2: Calculate the total number of ways to select one class in the morning and one in the afternoon.
- Since there are 4 choices in the morning and 3 choices in the afternoon, the total number of basic events is $n = 4 \times 3 = 12$.

Step 3: Identify the subjects related to arts and sciences.
- The subjects related to arts and sciences are politics, history, and geography.

Step 4: Calculate the number of ways to select at least one subject related to arts and sciences.
- In the morning, there are 2 classes related to arts and sciences (politics and geography), and in the afternoon, there is 1 class related to arts and sciences (history).
- The number of ways to select one class related to arts and sciences in the morning and any class in the afternoon is ${C}_{2}^{1}{C}_{3}^{1}$.
- The number of ways to select one class not related to arts and sciences in the morning and the class related to arts and sciences in the afternoon is ${C}_{2}^{1}{C}_{1}^{1}$.
- Therefore, the total number of basic events that include at least one subject related to arts and sciences is $m = {C}_{2}^{1}{C}_{3}^{1} + {C}_{2}^{1}{C}_{1}^{1} = 2 \times 3 + 2 \times 1 = 8$.

Step 5: Calculate the probability.
- The probability that at least one of the two selected classes is a subject related to arts and sciences is $p = \frac{m}{n} = \frac{8}{12} = \frac{2}{3}$.

Therefore, the correct answer is $\boxed{C}$, which corresponds to a probability of $\frac{2}{3}$.