Whenever you solve a probability question involving two conditions, and you are being asked to find the probability that either will occur for a given action, you are looking for what is known as a "union probability". Formally speaking, if we say A represents "the card is a Spade", and B represents "the card is a Queen", then we are looking for the probability of "the card is a Spade or a Queen", or symbolically:
P(A uu B)
The trick here is that these two possible events are not disjoint events; in other words, it can be possible to pull a single card and have it be a Spade and a Queen at the same time. The formula for determining P(A uu B) takes this into consideration:
P(A uu B) = P(A) + P(B) - P(A nn B)
(This is read as "the probability of A union B is equal to the probability of A plus the probability of B minus the probability of the intersection of A and B".)
If we consider P(A) (the probability the card is a Spade), in a standard deck of 52 cards there are exactly 13 cards which are Spades. Thus, P(A) = 13/52 = 1/4. (This is intuitive, because there are 4 suits of cards with the same values/ranks within them and we're only interested in one of those four suits.)
If we consider P(B) (the probability the card is a Queen), in a standard deck of 52 cards there are exactly 4 cards which are Queens (in suits of Hearts, Spades, Clubs, and Diamonds). Thus, P(B) = 4/52 = 1/13. (Again, this is intuitive, because there are 13 unique values of cards, of which there is only one Queen value.)
However, the probability P(A nn B) represents the probability the card is a Spade and a Queen at the same time. Of all 52 cards in the deck, there is only one Queen of Spades, thus P(A nn B) = 1/52.
Thus:
P(A uu B) = P(A) + P(B) - P(A nn B)
= 13/52 + 4/52 - 1/52 = 16/52 = 4/13