Question
Download Solution PDFIf P(A) = 0.5, P(B) = 0.7 and P(A ∩ B) = 0.3, then what is the value of P(A' ∩ B') + P(A' ∩ B) + P(A ∩ B') ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
- P(A') = 1 - P(A)
- P(A' ∩ B') = P((A ∪ B)') {Probability of not A and not B}
- P(A and not B) = P(A ∩ B') = P(A) - P(A ∩ B)
- P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
Calculation:
Given: P(A) = 0.5, P(B) = 0.7 and P(A ∩ B) = 0.3,
⇒ P(A ∪ B) = P(A) + P(B) - P(A ∩ B)
⇒ P(A ∪ B) = 0.5 + 0.7 - 0.3 = 0.9__(i)
Now the probability of not A and not B,
P(A' ∩ B') = P((A ∪ B)') = 1 - P(A ∪ B)
⇒ P(A' ∩ B') = 1 - 0.9 { From (i)}
⇒ P(A' ∩ B') = 0.1 __(ii)
The probability of B and not A,
P(A' ∩ B) = P(B) - P(A ∩ B)
⇒ P(A' ∩ B) = 0.7 - 0.3 = 0.4 __(iii)
The probability of A and not B,
P(A ∩ B') = P(A) - P(A ∩ B)
⇒ P(A' ∩ B) = 0.5 - 0.3 = 0.2 __(iv)
From (ii), (iii) and (iv),
P(A' ∩ B') + P(A' ∩ B) + P(A ∩ B') = 0.1 + 0.4 + 0.2 = 0.7
∴ The correct option is (2).
Last updated on May 30, 2025
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