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.
$P(Queen\cup \mathrm{♠})=P(Queen)+P(\mathrm{♠})$
We can draw a card from our two card deck. We are guaranteed to draw a spade, so $P(\mathrm{♠})=1$. Our chance of drawing a queen is 1 in 2, or $\frac{1}{2}$. Let's add everything together. $P(Queen\cup \mathrm{♠})=1.5$ or 150%. This obviously doesn't make sense.

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