What is the difference between independent and mutually exclusive events? Two events are mutually exclusive if they can't both happen.
Independent events are events where knowledge of the probability of one doesn't change the probability of the other.
Are these definitions correct? If possible, please give more than one example and counterexample. 
 A: If I toss a coin twice, the result of the first toss and the second toss are independent.
However the event that you get two heads is mutually exclusive to the event that you get two tails.
Suppose two events have a non-zero chance of occurring.
Then if the two events are mutually exclusive, they can not be independent.
If two events are independent, they cannot be mutually exclusive.
A: This question already has very good answers, I'm gonna add a visualization for independents events using some special diagrams. In these diagrams proportion of events to sample space represents their probability. Our sample space is a rectangle of 9x5 = 45 units:

We have event A (3x5) so P(A) = 3x5/9x5 = 15/45 = 1/3:

And event B (9x3) so P(B) = 9x3/9x5 = 27/45 = 3/5:

These two events intersect as:

∩ occupies 3x3 units:

(∣) = (∩) / () so (∣) = 9/27 = 1/3. But this is same as P(A)!

and (∣) = (∩) / () so (∣) = 9/15 = 3/5. But this is same as P()!

As in the two last diagrams, occurrence of one event doesn't affect the probability of the other event, these two events are called independent. So knowledge about occurrence of one of them doesn't affect our knowledge about probability of the other one. But this is not because they have nothing in common, on the contrary they are kinda in harmony by wiping out (the given event reduces sample space to itself, so it wipes out its complement) sample space in such a way that the other event proportion to given event doesn't change. I'd like to remember them as perpendicular events.
A: After reading the answers above I still could not understand clearly the difference between mutually exclusive AND independent events. I found a nice answer from Dr. Pete posted on math forum. So I attach it here so that op and many other confused guys like me could save some of their time.

If two events A and B are independent a real-life example is the following.  Consider a fair coin and a fair
  six-sided die.  Let event A be obtaining heads, and event B be rolling
  a 6.  Then we can reasonably assume that events A and B are 
  independent, because the outcome of one does not affect the outcome of
  the other.  The probability that both A and B occur is
P(A and B) = P(A)P(B) = (1/2)(1/6) = 1/12.
An example of a mutually exclusive event is the following.  Consider a
  fair six-sided die as before, only in addition to the numbers 1 
  through 6 on each face, we have the property that the even-numbered
  faces are colored red, and the odd-numbered faces are colored green.
  Let event A be rolling a green face, and event B be rolling a 6.  Then
P(B) = 1/6
P(A) = 1/2
as in our previous example.  But it is obvious that events A and B
  cannot simultaneously occur, since rolling a 6 means the face is red,
  and rolling a green face means the number showing is odd.  Therefore
P(A and B) = 0.
Therefore, we see that a mutually exclusive pair of nontrivial events
  are also necessarily dependent events.  This makes sense because if A
  and B are mutually exclusive, then if A occurs, then B cannot also
  occur; and vice versa.  This stands in contrast to saying the outcome
  of A does not affect the outcome of B, which is independence of
  events.

A: Mutually exclusive event :- two events are mutually exclusive event when they cannot occur at the same time. e.g if we flip a coin it can only show a head OR a tail, not both.
Independent event :- the  occurrence of one event does not affect the  occurrence of the others
e.g if we flip a coin two times, the first time may show a head, but this does not guarantee that the next time when we flip the coin the outcome will also be heads. From this example we can see the first event does not affect the occurrence of the next event.
A: Yes, that's fine.
Events are mutually exclusive if the occurrence of one event excludes the occurrence of the other(s).  Mutually exclusive events cannot happen at the same time. For example: when tossing a coin, the result can either be heads or tails but cannot be both.
$$\left.\begin{align}P(A\cap B) &= 0 \\ P(A\cup B) &= P(A)+P(B)\\ P(A\mid B)&=0 \\ P(A\mid \neg B) &= \frac{P(A)}{1-P(B)}\end{align}\right\}\text{ mutually exclusive }A,B$$
Events are independent if the occurrence of one event does not influence (and is not influenced by) the occurrence of the other(s).  For example: when tossing two coins, the result of one flip does not affect the result of the other.
$$\left.\begin{align}P(A\cap B) &= P(A)P(B) \\ P(A\cup B) &= P(A)+P(B)-P(A)P(B)\\ P(A\mid B)&=P(A) \\ P(A\mid \neg B) &= P(A)\end{align}\right\}\text{ independent }A,B$$
This of course means mutually exclusive events are not independent, and independent events cannot be mutually exclusive.  (Events of measure zero excepted.)
A: Events are Independent when happening of one does not influence happening of other. Eruption of volcano on Earth and orbit of Mars do not influence each other, so are independent events.
Growth of human population and preservation of many other species are mutually exclusive, as the one can only happen if the other does not happen.
Strictly speaking, mutually exclusive does not imply that one of them must happen. If there is a large asteroid impact on Earth, then neither human population grows nor endangered species are preserved.
A: Let $A$ be the event of hitting a dartboard's bullseye, and $B$ be the event of hitting within its inner ring.

*

*

Are these definitions correct?

Two events are mutually exclusive if they can't both happen.


False: Events $A$ and $B$ are mutually exclusive $\Big(P(A\cap
B)=P(A)=0\Big),$ yet do happen simultaneously.
Definition: Two events are mutually exclusive if they almost
never or never both happen, that is, if their joint probability is
zero.


*


Independent events are events where knowledge of the probability of one doesn't change the probability of the other.


False: Events $A$ and $B$ are independent $\Big(P(A\cap
B)=P(A)=0=P(A)\,P(B)\Big),$ yet the knowledge that the probability of $A$ is $1$ (i.e., that it occurs) increases $B$'s probability to $1.$
Informal characterisation: Two events, at least one of which has a
positive probability, are independent if the knowledge that one
happens doesn't change the other's probability.
Definition: Events $X$ and $Y$ are independent if $\,P(X\cap
Y)=P(X)\,P(Y).$


*

Is there any connection between independent events and mutually exclusive events?

Let $X$ and $Y$ be positive-probability events. If $X$ and $Y$ are mutually exclusive, then $P(X\cap Y)=0;$ since $P(X)\,P(Y)>0,$ we
have that $P(X\cap Y)\ne P(X)\,P(Y),$ that is, $X$ and $Y$ are
dependent.
Intuitively: if $X$ and $Y$ are mutually exclusive, then they almost never or never both happen; thus, knowing that event $Y$
happens means knowing that event $X$ almost doesn't or doesn't
happen; so, knowing that $Y$ happens decreases $X$'s probability to
zero; that is, events $X$  and $Y$ are dependent.
Be careful though: events $A$ and $B$ above are mutually exclusive
but independent, while the events {landing $'3'$ in a die roll}
and {landing an odd number in a die roll} are dependent although
not mutually exclusive.
A: It will be easier if we distinguish "mutually exclusiveness" from "independency" by considering the sample space in mind.

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*Two events that are compared for mutually exclusiveness must be from a single sample space. For example,

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*Tossing a coin twice.   $A=\{HH\}$ is an event in which the head shows up twice and $B=\{TT\}$ is an event in which the tail shows up twice. Their share the same sample space $S=\{HH,HT,TH,TT\}$. As $A\cap B=\{\}$, they are mutually exclusive.



*Two events that are compared for independency must be from  two sample spaces. For example,

*

*Tossing a coin twice.   $A=\{H\}$ is an event in which the head shows up in the first throw and  $B=\{T\}$ is an event in which the tail shows up in the second throw. The sample space for the first trial is $S_1=\{H,T\}$ and the sample space for the second trial is $S_2=\{H,T\}$ As $S_1=S_2$, they are independent.



The next additional questions are "is it possible to have 2 events that are"

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*"both mutually exclusive and independent?"


*"both mutually exclusive and dependent?"
I will update this answer in the future to answer the additional questions above.
