36 exam-style questions on Venn/tree/sample space; combined events; conditional; independence — topic 4.6 of Statistics and Probability in the IB Mathematics: Applications and Interpretation syllabus — with answers and explanations. Samples below; the full set is free with an account.
In a class of 30 students, 18 study Physics, 14 study Chemistry, and 8 study both. Determine how many students study neither subject.
Answer: 6
By inclusion–exclusion, students studying at least one = 18 + 14 − 8 = 24, so 30 − 24 = 6 study neither.
A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. Calculate the probability that both are red.
Answer: 5/14
Without replacement: P(both red) = (5/8) × (4/7) = 20/56 = 5/14.
For events A and B, P(A) = 0.6, P(B) = 0.5 and P(A ∪ B) = 0.8. Determine P(A ∩ B).
Answer: 0.3
P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.6 + 0.5 − 0.8 = 0.3.
A coin is biased so that P(heads) = 0.7. It is tossed twice. Calculate the probability of getting exactly one head.
Answer: 0.42
Exactly one head = HT or TH = (0.7)(0.3) + (0.3)(0.7) = 0.21 + 0.21 = 0.42.
Given P(A) = 0.5, P(B) = 0.4 and P(A | B) = 0.5, identify the correct statement about events A and B.
Answer: Independent; P(A ∩ B) = 0.20
P(A | B) = P(A) = 0.5, so A and B are independent; P(A ∩ B) = P(A | B) × P(B) = 0.5 × 0.4 = 0.20.
A disease affects 1% of a population. A test gives a positive result for 90% of people who have the disease and a negative result for 90% of people who do not. A person tests positive. Identify the numerator of P(disease | positive).
Answer: 0.90 × 0.01
P(disease | positive) = P(disease and positive) / P(positive); the numerator is P(disease) × P(positive | disease) = 0.01 × 0.90. The product 0.99 × 0.10 is the false-positive term, which belongs only in the denominator.
That was 6 of 36 Venn/tree/sample space; combined events; conditional; independence questions.
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36 Venn/tree/sample space; combined events; conditional; independence questions — free with an account. Spaced repetition, streaks and full exam simulations included.
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