Check whether the following probabilities P(A) and P(B) are consistent

1.	Check whether the following probabilities P(A) and P(B) are consistently defined, (i) P(A) = 0.5, P(B) = 0.7, P(A∩B) = 0.6 (ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ P) = 0.8 Note: P(A) and P(B) P(A ∩ B) are consistently defined iff P(A∩B) ≤ P(A) and P(B)
Answer» (i) P(A) = 0.6 and P(B) = 0.7 and P(A ∩ B) = 0.6 Here, p(A n B) > P(A) and P(A ∩ B)< P(B) P(A ∩ B) is not less than or equal to P(A) and P(B) ∴ P(A) and P(B), P(A n B) are not consistently defined. (ii) We have, P(A ∪ B) = P(A) + P(B) – p(A ∩ B) P(A ∩ B) = P(A) + P(B) – P(A ∪ B) = 0.5 + 0.5 – 0.8 = 0.9 – 0.8 = 0.1 ∴ P(A ∩ S) = 01 Here, P(A ∩ B) is less than P(A) and P(B) P(A) and P(B)P(A ∪ B) are consistently defined. (Note: P(A ∪ B)<P(A) + P(B))

Check whether the following probabilities P(A) and P(B) are consistently defined, (i) P(A) = 0.5, P(B) = 0.7, P(A∩B) = 0.6 (ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ P) = 0.8 Note: P(A) and P(B) P(A ∩ B) are consistently defined iff P(A∩B) ≤ P(A) and P(B)

Answer»

(i) P(A) = 0.6 and P(B) = 0.7 and P(A ∩ B) = 0.6

Here, p(A n B) > P(A) and P(A ∩ B)< P(B)

P(A ∩ B) is not less than or equal to P(A) and P(B)

∴ P(A) and P(B), P(A n B) are not consistently defined.

(ii) We have, P(A ∪ B) = P(A) + P(B) – p(A ∩ B)

P(A ∩ B) = P(A) + P(B) – P(A ∪ B)

= 0.5 + 0.5 – 0.8

= 0.9 – 0.8 = 0.1

∴ P(A ∩ S) = 01

Here, P(A ∩ B) is less than P(A) and P(B)

P(A) and P(B)P(A ∪ B) are consistently defined.

(Note: P(A ∪ B)<P(A) + P(B))

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Check whether the following probabilities P(A) and P(B) are consistently defined, (i) P(A) = 0.5, P(B) = 0.7, P(A∩B) = 0.6 (ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ P) = 0.8 Note: P(A) and P(B) P(A ∩ B) are consistently defined iff P(A∩B) ≤ P(A) and P(B)

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