We can solve this problem using the principle of inclusion-exclusion to find how many students play both basketball and baseball.

Let:

• $B$ be the set of students who play basketball.
• $A$ be the set of students who play baseball.
• $N$ be the total number of students, which is 25.

We are given the following information:

• $|B| = 11$ (11 students play basketball),
• $|A| = 5$ (5 students play baseball),
• 12 students play neither sport.

Step 1: Calculate the number of students who play at least one sport.

Since 12 students play neither basketball nor baseball, the number of students who play at least one sport is: $\text{Students playing at least one sport} = 25 - 12 = 13$

Step 2: Use the inclusion-exclusion principle.

The inclusion-exclusion principle states that: $|B \cup A| = |B| + |A| - |B \cap A|$ where $|B \cup A|$ is the number of students playing at least one sport, and $|B \cap A|$ is the number of students who play both sports.

We know that $|B \cup A| = 13$, so we can plug in the values: $13 = 11 + 5 - |B \cap A|$ Solving for $|B \cap A|$: $|B \cap A| = 16 - 13 = 3$

Thus, 3 students play both basketball and baseball.

Step 3: Calculate the probability.

The probability that a randomly chosen student plays both basketball and baseball is: $P(\text{both sports}) = \frac{|B \cap A|}{N} = \frac{3}{25}$

Final Answer:

The probability that a randomly chosen student plays both basketball and baseball is $\frac{3}{25}$.

Would you like any more details or explanations on this problem? Here are 5 related questions to explore:

1. What is the probability that a randomly chosen student plays only basketball?
2. How would the problem change if 15 students played neither sport?
3. What if we added a third sport, soccer, to the mix?
4. How can Venn diagrams help in visualizing problems like this?
5. What is the probability that a student chosen randomly plays only baseball?

Tip: In probability problems, drawing a diagram (like a Venn diagram) can often make it easier to visualize overlapping groups and apply inclusion-exclusion correctly.