To solve this problem, we can use the principle of inclusion-exclusion and the total number of students to calculate the probability.

Step 1: Understand the given information

• Total students: $29$
• Students who play basketball ($B$): $5$
• Students who play baseball ($A$): $19$
• Students who play neither: $7$

Step 2: Use the principle of inclusion-exclusion

We know that: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ Here:

• $n(A \cup B)$ is the number of students who play either basketball or baseball or both.
• $n(A \cap B)$ is the number of students who play both sports.

Students who play neither sport are excluded from $n(A \cup B)$, so: $n(A \cup B) = \text{Total students} - \text{Students who play neither}$ $n(A \cup B) = 29 - 7 = 22$

Now, substitute $n(A \cup B) = 22$, $n(A) = 19$, and $n(B) = 5$ into the inclusion-exclusion formula: $22 = 19 + 5 - n(A \cap B)$

Step 3: Solve for $n(A \cap B)$

$n(A \cap B) = 19 + 5 - 22 = 2$

So, $2$ students play both basketball and baseball.

Step 4: Calculate the probability

The probability that a randomly chosen student plays both basketball and baseball is: $P(\text{both}) = \frac{\text{Number of students who play both}}{\text{Total number of students}}$ $P(\text{both}) = \frac{2}{29}$

Final Answer:

The probability that a randomly chosen student plays both basketball and baseball is $\frac{2}{29}$ or approximately $0.069$ (6.9%).

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Would you like a deeper explanation or help with related problems? Here are 5 related questions for practice:

1. What is the probability that a randomly chosen student plays only baseball?
2. What is the probability that a randomly chosen student plays basketball or baseball but not both?
3. How many students play only basketball?
4. What is the probability that a randomly chosen student plays neither basketball nor baseball?
5. If 3 students are randomly chosen, what is the probability that at least one of them plays basketball?

Tip: Always verify your calculations using diagrams, such as a Venn diagram, to visualize overlapping sets.