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$\text{P}(A \cup B) = 0.7$ and $\text{P}(A\cup B^C) = 0.9.$ Determine $\text{P}(A).$

I have $\text{P}((A\cup B)^C) = 0.3$ and $\text{P}((A\cup B^C)^C) = 0.1$ and then I don't know what to do... I tried looking at the union, but I got confused trying to intersect and union everything together.

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Your calculation is helpful, since intersections are more convenient when we are using a Venn diagram. So draw the usual two intersecting ovals, labelling them $A$ and $B$.

We have $(A\cup B)^c=A^c\cap B^c$. So write $0.3$ in the region which is outside both $A$ and $B$.

We have $(A\cup B^c)^c=A^c\cap B$. So write $0.1$ in the region which is outside $A$ but in $B$.

Together, the two regions you have put numbers into cover everything outside $A$. Thus $\Pr(A)=1-0.3-0.1$.

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$P((A\cup B^C)^C) = 0.1 \implies P(A) = P(A \cup B) - P((A\cup B^C)^C) = 0.6$. You can see this from a Venn diagram. $(A\cup B^C)^C$ represents what is in $B$ but not in $A$.

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