Final answer:
To find the probability that a student took mathematics or history, did not take either subject, or only took history but not mathematics, we use the principles of set theory to determine the union and complement of the two sets. The probabilities calculated are 0.88 for taking mathematics or history, 0.12 for not taking either, and 0.34 for taking only history.
Step-by-step explanation:
To solve the question presented by the student regarding probability in a high school graduating class, we employ principles of set theory and combinatorics. The total number of students is 100. When considering events A (studied mathematics) and B (studied history), we use the formula to find the union of two sets: n(A ∪ B) = n(A) + n(B) - n(A ∩ B), where n(A) is the number of students who studied mathematics, n(B) who studied history, and n(A ∩ B) who studied both.
(a) The probability that a randomly selected student took mathematics or history is found by dividing the number of students who took at least one of these subjects by the total number of students. Given that 54 studied mathematics, 69 studied history, and 35 studied both, we have:
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(b) The probability that a student did not take either of these subjects is the complement of the probability that they took mathematics or history, which is 1 - Probability(A or B):
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(c) To find the probability that a student took history but not mathematics, we subtract those who took both from those who took history:
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