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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