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.