Basic probability question- multiplication rule I came across the following question in a textbook (bear in mind that this is the only information given)-
There is a $50$ percent chance of rain today.  There is a $60$ percent chance of rain tomorrow. There is a $30$ percent chance that it will not rain either day. What is the chance that it will rain both today and tomorrow? 
My instinct was to multiply the probabilities together for today and tomorrow to arrive at an answer of $30$ percent. However, the answer given is $40$ percent (based on taking the addition of the individual probabilities and subtracting their union). Can someone explain to me why the multiplication rule does not apply here? Does it have to do with independence? Bear in mind, I am trying to relearn probability theory from scratch. Thanks. 

Yes, the multiplication rule only applies when events are independent, and there is no reason to assume the events of rain on each day are independent. 
For this type of problem, it helps to make a Venn diagram. There are two circles, $A$ and $B$, representing the events of rain today and tomorrow. This gives four regions: $A\cap B$ (rain on both days), $A^c\cap B^c$ (no on both days), $A\cap B^c$ (rain today, but not tomorrow), and $A^c\cap B$. The given information, and the fact that the total probability is $1$, tells us that
\begin{align}
P(A)=P(A\cap B)+P(A\cap B^c)&=0.5\\
P(B)=P(A\cap B)+P(A^c\cap B)&=0.6\\
P(A^c\cap B^c)&=0.3\\
P(A\cap B)+P(A\cap B^c)+P(A^c\cap B)+P(A^c\cap B^c)&=1
\end{align}
This is four equations equations in four unknowns, allowing you to solve for $P(A\cap B)$.
