I'm not sure that I totally understand independent events. If the cards are the same suit or rank, then they have a 13/52 and 4/52 probability, respectively. However, I'm not totally sure how these two events relate to one another. I believe that they are independent.

7The are not independent, since they cannot both happen at the same time. – vadim123 May 06 '15 at 02:45
4 Answers
Mathematically, two events $A$ and $B$ with probabilities $P(A)$ and $P(B)$ are independent if
$$P(A \mid B) = P(A);$$ that is, if the probability of event $A$ happening, given that $B$ happened, is the same as the probability of $A$ happening with no other information.
We don't even need any computations for your problem. If we pick up two cards and they're the same suit, could they possibly have the same value?
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Mutual Exclusion and Independence seem to cause beginners a lot of unnecessary confusion. They are more or less on the opposite ends of a spectrum.
Events are independent if they have no influence on each others' occurrence. The measured probability of one event happening is not affected by knowledge that the other event happened. The conditional probability is $P(A\mid B) = P(A)$, which leads to the joint probability being $P(A\cap B) = P(A)P(B)$. (This is the Product Rule for Independent Events.)
Events are mutually exclusive if they influence each other to such an extent that they cannot occur simultaneously. The knowledge that one event happens completely affects the measure of the probability that the other event happened; it could not. The conditional probability is $P(A\mid B)=0$, which leads to the joint probability being $P(A\cap B)=0$.
For mutually exclusive events, the Additive Rule is $P(A\cup B) = P(A)+P(B)$.
For independent events the probability of the union is $P(A\cup B) = P(A)+P(B)P(A)P(B)$.
In this case, since two cards drawn from a standard deck cannot simultaneously have both the same value and the same suit, the events are mutually exclusive.
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The definition of independence for two events is $P(A\cap B) = P(A)P(B)$. Regardless of what something seems, if you can achieve the probability of the two events this way, they are independent. Having said that, does knowing one give you any information regarding the other? If not, then they are likely independent. This is shown using conditional probability: $P(A \mid B) = P(A)$.
What you are dealing with here is slightly different. The two events are mutually exclusive. Like the commenter said, they cannot occur at the same time. Mutually exclusive events are described by $P(A\cap B) = 0$, which is a little different.
That is discussed here What is the difference between independent and mutually exclusive events?
As is so often the case in stats, the formal definitions and equations get in the way of meaningful and intuitive understanding for beginners. So here's a simple explanation without equations that should make intuitive sense.
 We say that two events are "independent" of each other if the occurrence of one has no impact on the occurrence of the other.
 We say that two events are "mutually exclusive" of each other (also called "disjoint") if they can't both occur at the same time.
If you think about it, you can see that mutual exclusivity is a special type of nonindependence. That is, we have mutual exclusivity when the occurrence of one event impacts the other event in such a way as to reduce the likelihood of the other event to zero.
Here, let's say we've picked two cards from a deck. Event A is those cards having the same value, and Event B is those cards having the same suit. The two events are not independent, since the occurrence of Event A will have an impact on the occurrence of Event B. In fact, since Event B can't happen if Event A does (and vice versa), the occurrence of one reduces the likelihood of the other to zero, so we have an example of our special case of nonindependence that we call mutual exclusivity.