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.