What is the difference between independent and mutually exclusive events?
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.)
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.
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.