Explanation
Concept:
\textbf{Probability: The probability of the occurrence of an event A out of a total possible outcomes N, is given by: } P( A) = Nn(A), where n( A) is the number of ways in which the event A can oceur.
For two events A and B, we have: P(A U B)=P(A)+P(B)-P(A ∩B).
∙∙ If A and B are mutually exclusive events, then P(A ∩ B)=0.∙ If A and B are mutually exclusive events, then P(A ∩ B)=0.amp;If A and B are independent events, then P(A ∩ B)=P(A)×P(B).
· The complementary event of an event A is also denoted as Acor A'or A.
P(A)=1−P(A).
Set Theory: n( Ac∩Bc) = n( A∪B)c.
Calculation:
Let's look at the answer options one-by-one:
1) If P(U)=P(V), then n(U)=n(V), but that doesn't mean that U=V. Hence, FALSE.
2)If P(U)=0,then P(Uc)=1−0=1.It means that Uc=S. Hence, TRUE
3) If U ∩V=ф, then U and V are mutually exclusive, NOT independent. Hence, FALSE.
4) If U and V are independent, then P(U∩V)=P(U)×P(V) but P(U∁∩V∁)=P(U∪V)∁=P(U∁)×P(V∁).
Hence, FALSE.
