We are given the following information:

• Total number of students: $30$
• Students who play basketball: $12$
• Students who play baseball: $14$
• Students who play neither sport: $7$

Let's break this down step by step.

Step 1: Determine the number of students who play either basketball or baseball

Since 7 students play neither sport, the number of students who play at least one sport is:

$30 - 7 = 23$

Step 2: Use the principle of inclusion-exclusion

Let:

• $B$ be the set of students who play basketball.
• $A$ be the set of students who play baseball.

The total number of students who play either basketball or baseball can be expressed as:

$|B \cup A| = |B| + |A| - |B \cap A|$

Where:

• $|B| = 12$ (students who play basketball),
• $|A| = 14$ (students who play baseball),
• $|B \cup A| = 23$ (students who play either sport).

Plugging these values into the equation:

$23 = 12 + 14 - |B \cap A|$

Solving for $|B \cap A|$, which represents the number of students who play both basketball and baseball:

$|B \cap A| = 12 + 14 - 23 = 3$

So, 3 students play both basketball and baseball.

Step 3: Calculate the probability

The probability that a student chosen randomly from the class plays both sports is:

$P(\text{both}) = \frac{\text{Number of students who play both sports}}{\text{Total number of students}} = \frac{3}{30} = \frac{1}{10}$

Final Answer:

The probability that a randomly chosen student plays both basketball and baseball is $\frac{1}{10}$.

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Would you like further details or have any questions about this problem?

Here are some related questions for further exploration:

1. How would the probability change if 4 students played both basketball and baseball?
2. What is the probability that a student plays either basketball or baseball?
3. How can the principle of inclusion-exclusion be applied to other real-life scenarios?
4. If 2 more students were added to the class, and they both played basketball, how would this affect the probability?
5. How can you use a Venn diagram to visually represent the situation?

Tip: Always start with the principle of inclusion-exclusion when dealing with overlapping sets. It helps account for double-counted elements!