1
$\begingroup$

Good day everyone,

I am currently looking at a self-help excercise which goes about exhibiting the following behavior which really puzzled me. \begin{equation} > P(D) = P(A \cap D) + P(B \cap D) + P(C \cap D) \end{equation} After drawing out the Venn diagram, it just doesnt seem to make sense.

enter image description here

The original question that prompted my question as follows:

enter image description here

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

That's true for your problem (not for the general case), because A, B and C have no intersection between themselves, and the union of those three sets is equal to the universe set. People pay in cash OR in credit card OR in debit card and there is no other option of payment.

So your drawing isn't right for this case, because the only possible intersections are those between D and each one of the three sets A, B and C.

Improve this answer
$\endgroup$
3
  • 2
    $\begingroup$ +1, but technically 'that's true only if' is false: it suffices that the intersections are of probability 0 and that the universe minus $A\cap B\cap C$ is of probability 0. In addition, the statement could be true 'by accident', e.g., let $P(D)=1,~A\subset D,~ P(A)=1/3,~A=B=C$. Suggestion: edit to remove 'only' from the beginning sentence. $\endgroup$
    – Juho Kokkala
    Commented May 5, 2014 at 12:42
  • $\begingroup$ "and if the union of those three sets is equal to the universe set." This part can't be quite right? I can certainly draw a Venn diagram where the first clause of your sentence is true, and the pre-fixed condition is true (imagine A, B an C flush, but not overlapping, D intersects all three, but nothing else, i.e if P$(D \cap (A \cup B \cup C)^{c})$ = 0 is also true. $\endgroup$
    – Alexis
    Commented May 5, 2014 at 12:57
  • $\begingroup$ One more reason to remove the 'only', which I already did. $\endgroup$
    – Jundiaius
    Commented May 5, 2014 at 13:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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