To solve this problem, we use the principle of inclusion and exclusion along with basic probability concepts. Let's denote:

• $M$: The set of students who studied Mathematics.
• $H$: The set of students who studied History.
• $|M| = 54$: Number of students who studied Mathematics.
• $|H| = 69$: Number of students who studied History.
• $|M \cap H| = 35$: Number of students who studied both Mathematics and History.
• Total students = 100.

Now we calculate the answers step-by-step.

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(a) Probability that the student took Mathematics or History

The number of students who took Mathematics or History is given by: $|M \cup H| = |M| + |H| - |M \cap H|$ $|M \cup H| = 54 + 69 - 35 = 88$

The probability is: $P(M \cup H) = \frac{|M \cup H|}{\text{Total students}} = \frac{88}{100} = 0.88$

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(b) Probability that the student didn’t take either subject

The number of students who didn’t take either subject is: $\text{Students not in } M \cup H = \text{Total students} - |M \cup H| = 100 - 88 = 12$

The probability is: $P(\text{Neither}) = \frac{12}{100} = 0.12$

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(c) Probability that the student took History but not Mathematics

The number of students who took History but not Mathematics is: $|H \setminus M| = |H| - |M \cap H| = 69 - 35 = 34$

The probability is: $P(H \setminus M) = \frac{|H \setminus M|}{\text{Total students}} = \frac{34}{100} = 0.34$

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(d) Probability that the student took Mathematics but not History

The number of students who took Mathematics but not History is: $|M \setminus H| = |M| - |M \cap H| = 54 - 35 = 19$

The probability is: $P(M \setminus H) = \frac{|M \setminus H|}{\text{Total students}} = \frac{19}{100} = 0.19$

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Final Results:

• (a) $P(M \cup H) = 0.88$
• (b) $P(\text{Neither}) = 0.12$
• (c) $P(H \setminus M) = 0.34$
• (d) $P(M \setminus H) = 0.19$

Would you like detailed step-by-step calculations or additional clarifications?

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Here are 5 relative questions you can explore:

1. How does the principle of inclusion and exclusion apply to overlapping sets?
2. Can you calculate probabilities when more than two sets are involved?
3. What is the relationship between set theory and probability theory?
4. How would the probabilities change if the total number of students was different?
5. What is the significance of complementary probabilities in real-life problems?

Tip: Always ensure your probabilities add up to 1, as this is a quick check for accuracy.