In a certain Algebra 2 class of 28 students, 21 of them play basketball and 20 of them play baseball. There are 5 students who play neither sport. What is the probability that a student chosen randomly from the class plays both basketball and baseball?


Let's define some variables to organize the information:

- Let \( n = 28 \) be the total number of students in the class.
- Let \( B_b \) be the number of students who play basketball, which is \( B_b = 21 \).
- Let \( B_s \) be the number of students who play baseball, which is \( B_s = 20 \).
- Let \( N \) be the number of students who play neither sport, which is \( N = 5 \).
- Let \( x \) be the number of students who play both basketball and baseball, which is what we need to find.

### Step 1: Find the number of students who play either basketball or baseball or both.
Since 5 students play neither sport, the number of students who play either basketball or baseball or both is:
\[
28 - 5 = 23
\]
So, 23 students are involved in either or both sports.

### Step 2: Use the inclusion-exclusion principle to find the number of students who play both sports.
The inclusion-exclusion principle states that:
\[
\text{(Students who play either sport)} = B_b + B_s - \text{(Students who play both sports)}
\]
Substitute the known values:
\[
23 = 21 + 20 - x
\]
Now solve for \( x \):
\[
23 = 41 - x
\]
\[
x = 41 - 23 = 18
\]

### Step 3: Calculate the probability.
The probability that a randomly chosen student plays both basketball and baseball is the ratio of the number of students who play both sports to the total number of students:
\[
P(\text{Both sports}) = \frac{x}{n} = \frac{18}{28} = \frac{9}{14}
\]

Thus, the probability that a student chosen randomly plays both basketball and baseball is \( \frac{9}{14} \).

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the probability that a randomly chosen student plays only basketball?
2. What is the probability that a randomly chosen student plays only baseball?
3. How many students play only basketball?
4. How many students play only baseball?
5. What is the probability that a student plays either basketball or baseball but not both?

**Tip:** The inclusion-exclusion principle is helpful for solving problems involving overlapping groups!

In a certain Algebra 2 class of 28 students, 21 of them play basketball and 20 of them play baseball. There are 5 students who play neither sport. What is the probability that a student chosen randomly from the class plays both basketball and baseball?

Find the probability that a randomly chosen student plays both basketball and baseball in a class of 28 students, where 21 play basketball, 20 play baseball, and 5 play neither sport. This problem involves set theory, probability, and the inclusion-exclusion principle, suitable for Grades 10-11.

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