Ex 16.3, 12 - Check whether P(A), P(B) are consistently

Ex 16.3, 12 Check whether the following probabilities P(A) and P(B) are consistently defined • P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6 P(A) & P(B) are consistently defined if • P(A ∩ B) < P(A) & P(A ∩ B) < P(B) • P(A ∪ B) > P(A) & P(A ∪ B) > P(B) Given P(A) = 0.5, P(B) = 0.7, P(A ∩ B) = 0.6 Here, P(A ∩ B) > P(A). Hence, P(A) and P(B) are not consistently defined. Ex 16.3, 12 Check whether the following probabilities P(A) and P(B) are consistently defined (ii) P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8 P(A) & P(B) are consistently defined if • P(A ∩ B) < P(A) & P(A ∩ B) < P(B) • P(A ∪ B) > P(A) & P(A ∪ B) > P(B) Given P(A) = 0.5, P(B) = 0.4, P(A ∪ B) = 0.8 Here P(A ∪ B) > P(A) & P(A ∪ B) > P(B) Finding P(A ∩ B) We know that P(A ∪ B) = P(A) + P(B) – P(A ∩ B) Putting values 0.8 = 0.5 + 0.4 – P(A ∩ B) 0.8 = 0.9 – P(A ∩ B) P(A ∩ B) = 0.9 – 0.8 P(A ∩ B) = 0.1 So, P(A ∩ B) < P(A) & P(A ∩ B) < P(B) Since both conditions are satisfied, Hence, P(A) and P(B) are consistently defined.