Suppose that A and B are two events. Draw a Venn diagram and use it to explain how P(~A ∩ ~B) can be computed if all you know are P(A), P(B), and P(A ∩ B).

Question

Suppose that A and B are two events. Draw a Venn diagram and use it to explain how P(~A ∩ ~B) can be computed if all you know are P(A), P(B), and P(A ∩ B).

anonymous · Answer

first you need to know what $A^c\cap B^c$ looks like

anonymous · Answer

I was thinking it was everything outside of A union B

anonymous · Answer

yes exactly
so if you know the probability of $A\cup B$ you can subtract that number from 1 to get your answer

anonymous · Answer

that's what I thought, but why do they mention P(A) and P(B)?

anonymous · Answer

because you are not told $P(A\cup B)$ you are told only $P(A),P(B), P(A\cap B)$

anonymous · Answer

so now the question is, if you know those three numbers, how do you find $P(A\cup B)$ ?

anonymous · Answer

oh I see all those add up to the union.

anonymous · Answer

no not quite

anonymous · Answer

1-[P(A)+P(B)-P(A∩B)]

anonymous · Answer

since when you add P(A) and P(B) you add P(A∩B) twice

anonymous · Answer

yes now you have it exactly

anonymous · Answer

ok that makes sense, thanks