1. An elementary school is offering 3 language classes: one in Spanish, one in French,
and one in German. The classes are open to any of the 100 students in the school. There
are 28 students in the Spanish class, 26 in the French class, and 16 in the German class.
There are 12 students who are in both Spanish and French, 4 who are in both Spanish and
German, and 6 who are in both French and German. In addition, there are 2 students
taking all 3 classes.
(a) If a student is chosen randomly, what is the probability that he or she is not in
any of the language classes?
(b) If a student is chosen randomly, what is the probability that he or she is taking
exactly one language class?
(c) If 2 students are chosen randomly, what is the probability that at least 1 is taking
a language class?

Let's solve each part of the problem step by step using the information provided and set theory.

### Given Data:
- Total number of students: \(100\)
- Students in Spanish (\(S\)): \(28\)
- Students in French (\(F\)): \(26\)
- Students in German (\(G\)): \(16\)
- Students in both Spanish and French (\(S \cap F\)): \(12\)
- Students in both Spanish and German (\(S \cap G\)): \(4\)
- Students in both French and German (\(F \cap G\)): \(6\)
- Students in all three classes (\(S \cap F \cap G\)): \(2\)

#### (a) Probability that a student is not in any of the language classes

We use the principle of inclusion-exclusion to find the total number of students enrolled in at least one class:

\[
|S \cup F \cup G| = |S| + |F| + |G| - |S \cap F| - |S \cap G| - |F \cap G| + |S \cap F \cap G|
\]

Substituting the values:

\[
|S \cup F \cup G| = 28 + 26 + 16 - 12 - 4 - 6 + 2 = 50
\]

So, \(50\) students are taking at least one class. The number of students not taking any language class is:

\[
100 - 50 = 50
\]

Therefore, the probability that a randomly chosen student is not in any of the language classes is:

\[
P(\text{not in any class}) = \frac{50}{100} = 0.5
\]

#### (b) Probability that a student is taking exactly one language class

To find the number of students taking exactly one language class, calculate:

- Students only in Spanish: \( |S| - (|S \cap F| + |S \cap G| - |S \cap F \cap G|) = 28 - (12 + 4 - 2) = 14 \)
- Students only in French: \( |F| - (|F \cap S| + |F \cap G| - |S \cap F \cap G|) = 26 - (12 + 6 - 2) = 10 \)
- Students only in German: \( |G| - (|G \cap S| + |G \cap F| - |S \cap F \cap G|) = 16 - (4 + 6 - 2) = 8 \)

Total number of students taking exactly one language class is:

\[
14 + 10 + 8 = 32
\]

So, the probability that a randomly chosen student is taking exactly one language class is:

\[
P(\text{exactly one class}) = \frac{32}{100} = 0.32
\]

#### (c) Probability that at least 1 of the 2 chosen students is taking a language class

First, find the probability that a randomly chosen student is not taking any language class, which is \(0.5\).

For 2 students chosen randomly, the probability that both are not taking any class is:

\[
P(\text{both not in any class}) = 0.5 \times 0.5 = 0.25
\]

The probability that at least one of the two chosen students is taking a language class is:

\[
P(\text{at least 1 in a class}) = 1 - P(\text{both not in any class}) = 1 - 0.25 = 0.75
\]

### Summary of Results:
(a) Probability that a student is not in any class: \(0.5\)

(b) Probability that a student is taking exactly one language class: \(0.32\)

(c) Probability that at least 1 of the 2 students is taking a language class: \(0.75\)

---

Would you like any more details on these calculations or have further questions?

### Related Questions:
1. How would the probabilities change if the number of students in each class were different?
2. What is the probability that a student is enrolled in at least two classes?
3. How would you calculate the probability if three students were chosen instead of two?
4. What is the probability that a randomly chosen student is enrolled in both Spanish and German but not French?
5. How would adding a fourth language class affect these probabilities?

**Tip:** The Principle of Inclusion-Exclusion is a powerful tool for finding the count of elements in unions of sets by accounting for overlaps.

An elementary school is offering 3 language classes: one in Spanish, one in French, and one in German. The classes are open to any of the 100 students in the school. There are 28 students in the Spanish class, 26 in the French class, and 16 in the German class. There are 12 students who are in both Spanish and French, 4 who are in both Spanish and German, and 6 who are in both French and German. In addition, there are 2 students taking all 3 classes. (a) If a student is chosen randomly, what is the probability that he or she is not in any of the language classes? (b) If a student is chosen randomly, what is the probability that he or she is taking exactly one language class? (c) If 2 students are chosen randomly, what is the probability that at least 1 is taking a language class?

Explore how to calculate the probability of students enrolled in language classes using set theory and the Inclusion-Exclusion Principle. This problem involves Spanish, French, and German classes with specific overlaps and helps calculate the probability of various outcomes for a student body of 100. Suitable for Grades 10-12.

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