To determine the probability that Vinay will have a male history teacher two years in a row, given that the probability of having a male history teacher is \( \frac{3}{8} \), you would multiply the probability of having a male teacher in the first year by the probability of having a male teacher in the second year. Since these events are independent (the outcome of one year does not affect the other), you can calculate it as follows:
\[ P(\text{male teacher in 1st year}) \times P(\text{male teacher in 2nd year}) = \left( \frac{3}{8} \right) \times \left( \frac{3}{8} \right) = \left( \frac{3}{8} \right)^2 \]
So, the probability that he will have a male history teacher two years in a row is:
\[ \left( \frac{3}{8} \right)^2 \]
Therefore, the correct answer is \( \frac{3}{8} \) squared or \( \left( \frac{3}{8} \right)^2 \).