Probability what is $P(AB)$?

Question

If $P(A) = 0.2$, and $P(B) = 0.3$, what is $P(AB)$?

Does this intersect just add them together?

Answer 1

Without further information, all you can say is that $0 \le P(AB) \le 0.2$.

Answer 2

If $A$ and $B$ are independent than $P(AB) = P(A)P(B)$, otherwise we can't say anything without more information.

Answer 3

You have $P(A+B) = P(A) + P(B) - P(AB)$ and so $P(AB) = P(A) + P(B) - P(A+B)$

If $A$ and $B$ are mutually exclusive, then $P(A+B)=P(A)+P(B)=0.5$, and so then $P(AB)=0$

If $A$ and $B$ in any way overlap, then $P(AB)>0$ and $P(A + B) < 0.5$

If $A$ and $B$ completely overlap, then $P(AB)=min(P(A),P(B))=0.2$, and $P(A+B)=max(P(A),P(B))=0.3$

So, $0 \le P(AB)\le 0.2$ and $0.3 \le P(A+B)\le 0.5$

If $A$ and $B$ are independent, then $P(AB) = P(A) \cdot P(B)=0.06$