Addition Theorem
Given two events with P(A) = 0.6 and P(B) = 0.5 inside a sample space, while their union has probability 0.8, what is the probability of both occurring together?
Select the correct option:
Solution
0.3
The addition theorem of probability states that P(A ∪ B) = P(A) + P(B) − P(A ∩ B), correcting for the double-counted overlap when two events are not mutually exclusive. This identity is a foundational JEE Advanced tool whenever union and individual probabilities are linked. Rearranging gives P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.6 + 0.5 − 0.8 = 0.3. Option 0.2 results from mistakenly computing P(A) + P(B) − 2P(A ∪ B) or a sign slip. Option 0.4 comes from using 0.6 + 0.5 − 0.7 with a misread union value. Option 0.1 arises from subtracting the smaller probability from the union incorrectly. The result rests directly on the inclusion–exclusion principle for two sets, the probabilistic mirror of |A ∪ B| = |A| + |B| − |A ∩ B|. Plausibility check: P(A ∩ B) = 0.3 cannot exceed min(P(A), P(B)) = 0.5 and is non-negative, and substituting back gives 0.6 + 0.5 − 0.3 = 0.8, matching the stated union, so the answer is internally consistent.
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About This Question
- Subject
- mathematics
- Chapter
- statistics and probability
- Topic
- addition theorem
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
0.3
The addition theorem of probability states that P(A ∪ B) = P(A) + P(B) − P(A ∩ B), correcting for the double-counted overlap when two events are not mutually exclusive. This identity is a foundational JEE Advanced tool whenever union and individual probabilities are linked. Rearranging gives P(A ∩ B) = P(A) + P(B) − P(A ∪ B) = 0.6 + 0.5 − 0.8 = 0.3. Option 0.2 results from mistakenly computing P(A) + P(B) − 2P(A ∪ B) or a sign slip. Option 0.4 comes from using 0.6 + 0.5 − 0.7 with a misread union value. Option 0.1 arises from subtracting the smaller probability from the union incorrectly. The result rests directly on the inclusion–exclusion principle for two sets, the probabilistic mirror of |A ∪ B| = |A| + |B| − |A ∩ B|. Plausibility check: P(A ∩ B) = 0.3 cannot exceed min(P(A), P(B)) = 0.5 and is non-negative, and substituting back gives 0.6 + 0.5 − 0.3 = 0.8, matching the stated union, so the answer is internally consistent.
This easy difficulty mathematics question is from the chapter statistics and probability, covering the topic of addition theorem. It appeared in the 2025 exam.
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