https://file.aiquickdraw.com/mathbot/file/1725474235388w7op9elo.jpg

Let's break down and solve the problem:

### Problem Statement:
- **Total students in the class (n)** = 80
- **Students studying Mathematics (M)** = 29
- **Students studying Physics (P)** = 12
- **Students studying both subjects (M ∩ P)** = 8

### Required Probability:
Find the probability that a randomly selected student **does not study both** Mathematics and Physics.

### Steps to Solve:

1. **Calculate the number of students studying only Mathematics (M only)**:
   \[
   M \text{ only} = M - (M \cap P) = 29 - 8 = 21
   \]

2. **Calculate the number of students studying only Physics (P only)**:
   \[
   P \text{ only} = P - (M \cap P) = 12 - 8 = 4
   \]

3. **Calculate the number of students studying either Mathematics or Physics but not both**:
   \[
   \text{Only Mathematics or Only Physics} = M \text{ only} + P \text{ only} = 21 + 4 = 25
   \]

4. **Calculate the number of students who do not study both subjects (including those who study only one subject or neither subject)**:
   - **Students who study both** (already given): \( M \cap P = 8 \)
   - **Students who study neither**: 
     \[
     \text{Neither} = n - (M + P - M \cap P) = 80 - (29 + 12 - 8) = 80 - 33 = 47
     \]

5. **Total students who do not study both subjects**:
   \[
   \text{Students not studying both} = n - M \cap P = 80 - 8 = 72
   \]

6. **Calculate the probability**:
   The probability that a randomly selected student does not study both subjects is:
   \[
   P(\text{not both}) = \frac{\text{Students not studying both}}{n} = \frac{72}{80} = \frac{9}{10} = 0.9
   \]

### Final Answer:
The probability that a randomly selected student does not study both Mathematics and Physics is **0.9** or **90%**.

Would you like more details on any part of this solution or have any further questions? Here are some related questions you might consider:

1. What is the probability that a student studies only Mathematics?
2. What is the probability that a student studies only Physics?
3. How many students study neither Mathematics nor Physics?
4. What is the probability that a student studies at least one of these subjects?
5. If another student is added to the class who studies both subjects, how does this affect the probabilities?

**Tip:** In probability problems involving sets, using Venn diagrams can help visualize and solve the problem more easily.

In a class of 80 IB students, 29 study mathematics; 12 study physics, and 8 study both of these subjects. Find the probability that a randomly selected student does not study both of these subjects.

Learn how to calculate the probability that a randomly selected student from a class of 80 does not study both mathematics and physics. This problem covers key concepts in probability, set theory, and basic algebra, suitable for students in Grades 9-12.

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