In a certain Algebra 2 class of 25 students, 11 of them play basketball and 5 of them play baseball. There are 12 students who play neither sport. What is the probability that a student chosen randomly from the class plays both basketball and baseball?

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.

In this Algebra 2 probability problem, calculate the probability that a randomly selected student plays both basketball and baseball using the inclusion-exclusion principle. With 11 basketball players, 5 baseball players, and 12 students who play neither sport, discover the solution step-by-step. Ideal for students in Grades 9-12.

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