What's the probability of drawing a red card or a card with a face (king, queen, or jack) from a standard deck of 52 cards?

Question

What's the probability of drawing a red card or a card with a face (king, queen, or jack) from a standard deck of 52 cards?

1. A. 13/15
2. B. 1/4
3. C. 29/52
4. D. 8/13 (Correct answer)

Correct Answer

Option D — 8/13

Detailed Solution & Explanation

The correct answer is 8/13.

Key Points

• A standard deck has 52 cards: 26 red (hearts, diamonds) and 26 black (spades, clubs), with 12 face cards (King, Queen, Jack in each of the 4 suits).
• This is an "or" question, so use the addition rule:
  • $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Counting each set:
  • Red cards: $n(A) = 26$
  • Face cards: $n(B) = 12$
  • Red face cards (counted in both): $n(A \cap B) = 6$ — King, Queen and Jack of hearts and of diamonds.
• Applying the rule:
  • $n(A \cup B) = 26 + 12 - 6 = 32$
  • $P = \frac{32}{52} = \frac{8}{13}$

Shortcut Trick

• Count directly instead: 26 red cards + the 6 black face cards (K, Q, J of spades and clubs) $= 32$. Adding only the black face cards avoids the overlap entirely, so no subtraction is needed.

Additional Information

• Forgetting to subtract the overlap gives $\frac{38}{52}$, the single most common error in "or" probability questions — the 6 red face cards would otherwise be counted twice.
• Deck facts worth memorising: 4 suits × 13 cards; 12 face cards (some conventions include Aces, giving 16 — read the question's own definition, which here specifies "king, queen, or jack"); 2 red suits and 2 black suits.
• Distractor check: $\frac{29}{52}$ comes from subtracting 9 instead of 6, and $\frac{1}{4}$ is simply the probability of a single suit.