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A standard deck has 52 cards: 4 suits (Hearts ♥, Diamonds ♦, Clubs ♣, Spades ♠) × 13 ranks. Face cards are Jack, Queen, King (12 total). Red cards are Hearts + Diamonds (26 total).
Can a card be BOTH a Heart AND a Face card?
Yes! The Jack, Queen, and King of Hearts are both Hearts and Face cards. These events OVERLAP, so we need the full inclusion-exclusion formula.
Why can't we just add P(Heart) + P(Face)?
If we add them directly, the J♥, Q♥, and K♥ get counted TWICE — once as Hearts and once as Face cards. We must subtract the overlap to avoid double-counting.
The Inclusion-Exclusion Formula:
Draw one card from a standard deck. What's P(Heart OR Face Card)?
J, Q, K in all four suits
These cards are counted in BOTH sets above. If we don't subtract them, they'll be double-counted!
The Venn diagram shows the overlap — 3 cards that are both Hearts and Face cards
Add the individual probabilities
This counts some cards twice (the J♥, Q♥, K♥)
Subtract the overlap to fix double-counting
Remove the 3 cards counted twice (J♥, Q♥, K♥)
Result:
Count: 13 hearts + 9 non-heart face cards = 22 cards
(We add 9 because 3 face cards are already counted as hearts)