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If A and B are independent events with P(A) = .5 and P(B) = .2, find the following:a) P(A U B)b) P(A^c ? B^c)c) P(A^c U B^c)**Note A^c or B^c refers to A complement or B complement**

Answer :

LammettHash

If A and B are independent, then [tex]P(A\cap B)=P(A)P(B)[/tex].

a.

[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)=P(A)+P(B)-P(A)P(B)[/tex]

[tex]P(A\cup B)=0.5+0.2-0.5\cdot0.2[/tex]

[tex]\boxed{P(A\cup B)=0.6}[/tex]

b. I'm guessing the ? is supposed to stand for intersection. We can use DeMorgan's law for complements here:

[tex]P(A^c\cap B^c)=P(A\cup B)^c=1-P(A\cup B)[/tex]

[tex]P(A^c\cap B^c)=1-0.6[/tex]

[tex]\boxed{P(A^c\cap B^c)=0.4}[/tex]

c. DeMorgan's law can be used here too:

[tex]P(A^c\cup B^c)=P(A\cap B)^c=1-P(A\cap B)=1-P(A)P(B)[/tex]

[tex]P(A^c\cup B^c)=1-0.5\cdot0.2[/tex]

[tex]\boxed{P(A^c\cup B^c)=0.9}[/tex]

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