What's the difference between Independent Event and Mutually Inclusive Event? What's the difference between Independent Event and Mutually Inclusive Event?
Are all Independent Events mutually inclusive or vice verse? 
 A: Events $A$ and $B$ are $\textbf{independent}$ if $A$ occurring does not affect the probability of $B$ occurring. More precisely, events $A$ and $B$ are $\textbf{independent}$ if $P(A\cap B)=P(A)\cdot P(B).$
Two events are $\textbf{mutually inclusive}$ if they can occur exactly at the same time. More precisely, events $A$ and $B$ are $\textbf{mutually inclusive}$ if $A\cap B\not=\varnothing$. That is, they share $\textit{common outcomes}$. 
$\textbf{Example of (two) independent events}$: suppose you toss a coin and a six-sided die. The probability of the coin landing on its tail is $\frac{1}{2}$ while the probability of the die landing on the number $6$ is $\frac{1}{6}$. Thus the probability of the coin landing on its tail and the die landing on the number $6$ is $\frac{1}{2} \cdot\frac{1}{6} =\frac{1}{12}$. 
$\textbf{Example of mutually inclusive events}$: suppose you roll a six-sided die wanting to get the number $6$ on the top. Then the number $1$ must be on the bottom. Thus, one cannot happen without the other occurring. 
$\textbf{Second example of mutually inclusive events}$: let $A$ be the event containing positive integers less than $5$ and let $B$ be the event containing even integers from $1$ to $10$. Then 
$$
\begin{align*}
A &= \{1,2,3,4 \}, \\ 
B &= \{2,4,6,8,10 \}. \\ 
\end{align*}
$$
Since there are common outcomes $A\cap B=\{ 2,4 \}$ between $A$ and $B$, we say $A$ and $B$ are mutually inclusive. 
