The problem I am working on is:
The three most popular options on a certain type of new car are a built-in GPS (A), a sunroof (B), and an automatic transmission (C). If 40% of all purchasers request A, 55% request B, 70% request C, 63% request A or B, 77% request A or C, 80% request B or C, and 85% request A or B or C, determine the probabilities of the following events. [Hint: “Aor B” is the event that at least one of the two options is requested; try drawing a Venn diagram and labeling all regions.]
a.The next purchaser will request at least one of the three options.
b.The next purchaser will select none of the three options.
c. The next purchaser will request only an automatic transmission and not either of the other two options.
d.The next purchaser will select exactly one of these three options.
I am absolutely positive that $P(A)=40\%$, $P(B)=55\%$, $P(C)=70\%$, and $P(A \cup B \cup C)=85\%$. However, the pieces of data I am not quite certain about are $P(A \cup B \cap C')=63\%$, $P(A \cup C \cap B')=77\%$, $P(B \cup C \cap A')=80\%$, do these values correspond to the rest of the data? If so, then I seems nearly impossible to be able to generate the Venn Diagram. Could someone help?
EDIT: What I am having a difficult time interpreting is, when they say in the question, "...63% request A or B." To me, that says only A or only B; and under this interpretation I would write $P(A \cup B \cap C')=63\%$. Under André Nicolas' interpretation, "63% request A or B," means $P(A \cup B)=63\%$. If it is the case that André Nicolas is correct, then it seems like they should have stated in the question, "63% request A or B, A and C, B and C, or A and B and C."
Also, I solved the problem under André Nicolas' assumption, and for part d), I know the answer but I am sure how to put in it math symbols. How would I do that?