Let Ω be the sample space and A⊆Ω be an event. Given below are two statements:(S1): If P(A)=0, then A=ϕ(S2): If P(A)=1, then A=ΩThenonly (S1) is trueonly (S2) is trueboth (S1) and (S2) are trueboth (S1) and (S2) are false
Question
Let Ω be the sample space and A⊆Ω be an event. Given below are two statements:(S1): If P(A)=0, then A=ϕ(S2): If P(A)=1, then A=ΩThenonly (S1) is trueonly (S2) is trueboth (S1) and (S2) are trueboth (S1) and (S2) are false
Solution
Both (S1) and (S2) are false.
Explanation:
(S1) is false because P(A)=0 does not necessarily mean that A is an empty set. It just means that the event A is not going to happen. There could be elements in A, but they may not be favorable outcomes, hence the probability is 0.
(S2) is also false because P(A)=1 does not necessarily mean that A=Ω. It just means that the event A is certain to happen. But it doesn't mean that A has to be the entire sample space. It could be a subset of the sample space that includes all the favorable outcomes.
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