Introduction to Statistics/Still confused?

Consider a simpler situation involving a deck of two cards: the queen of spades and the two of spades. We'll define our joint probability again as "the chance of drawing a queen or a spade", and naively express it as. $$P(Queen \cup \spadesuit) = P(Queen) + P(\spadesuit)$$ We can draw a card from our two card deck. We are guaranteed to draw a spade, so $$P(\spadesuit) = 1$$. Our chance of drawing a queen is 1 in 2, or $$\tfrac{1}{2}$$. Let's add everything together. $$P(Queen \cup \spadesuit) = 1.5$$ or 150%. This obviously doesn't make sense.

If we use the correct definition of the probability for this event: $$P(Queen \cup \spadesuit) = P(Queen) + P(\spadesuit) - P(Queen \cap \spadesuit)$$ $$ = 0.5 + 1.0 - 0.5$$ we get the answer we expect.