Let $Ω$ be any type of sample room, and $A, B$ are subsets of $Ω$. Which of the following statements are always true?

If $|A|+|B|≥|Ω|$, then $P(A∪B)=1$If $|A|+|B|≥|Ω|$, then $P(A)+P(B)≥1$If $P(A)+P(B)>1$, then $A∩B≠∅$If $P(A)+P(B)>1$, then $P(A∪B)=1$

I would certainly select alternatives 2 and 3, however I struggle to identify just how either among these have the right to organize true or false...

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(1) and (4) are false in general. Just take $Omega = 1,2,3$ and $jiyuushikan.orgbb P$ uniformly dispersed, take $A = B = 1,2$. Then $jiyuushikan.orgbb P(A cup B) = frac23$, however $|A|+|B| = 4 ge 3 = |Omega|$, and also $jiyuushikan.orgbb P(A) + jiyuushikan.orgbb P(B) = frac43 > 1$

(3) is true in basic. Keep in mind that $jiyuushikan.orgbb P(A cup B) = jiyuushikan.orgbb P(A) + jiyuushikan.orgbb P(B) - jiyuushikan.orgbb P(A cap B)$. If $jiyuushikan.orgbb P(A) + jiyuushikan.orgbb P(B) > 1$, then $jiyuushikan.orgbb P(A cap B) $ must be greater than $0$, too, because $jiyuushikan.orgbb P(A cup B) $ cannot be higher than $1$.

About the (2), I need to have actually more information, favor is $Omega$ finite, and also $jiyuushikan.orgbb P$ uniformly distributed? If so, then answer if yes. But not in general


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