Introduction to Statistics/Still confused?

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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.

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