To solve this problem, we will use the principle of inclusion and exclusion.

Step 1: Understanding the data

• Total number of students: 116
• Number of students in each language class:
  • Spanish: 45
  • French: 36
  • German: 28
• Students taking both languages:
  • Spanish and French: 17
  • Spanish and German: 8
  • French and German: 7
• Students taking all three languages: 4

Step 2: Inclusion-Exclusion Formula

The inclusion-exclusion principle helps us calculate the number of students taking at least one language class. The formula for the union of three sets (students in Spanish, French, or German) is:

$|S \cup F \cup G| = |S| + |F| + |G| - |S \cap F| - |S \cap G| - |F \cap G| + |S \cap F \cap G|$

Where:

• $|S|$ is the number of students in Spanish (45),
• $|F|$ is the number of students in French (36),
• $|G|$ is the number of students in German (28),
• $|S \cap F|$ is the number of students in both Spanish and French (17),
• $|S \cap G|$ is the number of students in both Spanish and German (8),
• $|F \cap G|$ is the number of students in both French and German (7),
• $|S \cap F \cap G|$ is the number of students in all three languages (4).

Step 3: Plug the values into the formula

$|S \cup F \cup G| = 45 + 36 + 28 - 17 - 8 - 7 + 4$

$|S \cup F \cup G| = 81$

This means 81 students are taking at least one language class.

Step 4: Calculate the probability

The probability that a randomly chosen student is taking at least one language class is the ratio of students taking at least one language to the total number of students in the school.

$P(\text{at least one class}) = \frac{\text{Students in at least one class}}{\text{Total students}} = \frac{81}{116}$

$P(\text{at least one class}) \approx 0.698$

Final Answer:

The probability that a randomly chosen student is taking at least one language class is approximately 0.698.

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Would you like more details on the inclusion-exclusion method or have any further questions?

Here are some related questions:

1. How would you apply inclusion-exclusion to four sets instead of three?
2. What is the probability that a student is taking exactly two language classes?
3. How would you adjust this calculation if some students were not taking any language classes?
4. What is the probability of a student taking only one language class?
5. Can you solve for the number of students in each pair of classes (e.g., Spanish and French but not German)?

Tip: The inclusion-exclusion principle helps simplify problems involving overlapping sets, making it much easier to calculate probabilities involving multiple categories!