Explanation
Given: A⊂B (Set A is a subset of Set B).
-
Set Representation: If A⊂B, then B consists of elements in A and elements that are in B but not in A. The region "in B but not in A" is represented as (Aˉ∩B). Therefore:
This part of the statement is True.
-
Probability Comparison: Since A⊂B, the number of outcomes in A is less than or equal to the number of outcomes in B. By the monotonicity property of probability:
The statement claims P(A) > P(B), which is False.
Conclusion: Statement I is False.
Evaluation of Statement II
Given: A and B are independent events, which means P(A∩B)=P(A)⋅P(B).
-
Theorem of Independence: It is a standard result in probability that if two events are independent, their complements and combinations with complements are also independent.
-
Verification for Aˉ and Bˉ:
P(Aˉ∩Bˉ)=1−[P(A)+P(B)−P(A)P(B)]
P(Aˉ∩Bˉ)=[1−P(A)][1−P(B)]=P(Aˉ)⋅P(Bˉ)
This confirms they are independent. The same logic applies to (Aˉ and B) and (A and Bˉ).
Conclusion: Statement II is True.
Final Answer
Based on the analysis above:
-
Statement I is False.
-
Statement II is True.
