To find the experimental probability of the next contestant qualifying for the bonus round, we can use the results from the previous contestants.
The formula for experimental probability is:
\[ P(\text{event}) = \frac{\text{number of successful outcomes}}{\text{total number of outcomes}} \]
In this case, the successful outcome is qualifying for the bonus round. From the last 12 contestants, 3 qualified. Therefore:
\[ P(\text{bonus round}) = \frac{3}{12} \]
This fraction can be simplified:
\[ P(\text{bonus round}) = \frac{1}{4} \]
So, the experimental probability that the next contestant will qualify for the bonus round is:
\[ \boxed{\frac{1}{4}} \]