Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$\text{P}(A \cup B) = 0.7$ 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.

share|improve this question
add comment

2 Answers 2

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 $\Pr(A)=1-0.3-0.1$.

share|improve this answer
add comment

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

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.