A card is drawn at random from a standard 52-card deck. What is the probability of drawing a club or a face card? Note: there are 3 face cards in each suit (the King, Queen, and Jack). Give your answer as a fraction.
The correct answer and explanation is:
To find the probability of drawing a club or a face card from a standard 52-card deck, we use the principle of inclusion and exclusion.
Step 1: Total cards in a deck
A standard deck has 52 cards.
Step 2: Count the number of clubs
There are 13 clubs (one for each rank: A, 2–10, J, Q, K).
Step 3: Count the number of face cards
Each suit has 3 face cards: Jack, Queen, and King.
There are 4 suits (clubs, diamonds, hearts, spades), so:
4 × 3 = 12 face cards
Step 4: Count the overlap (club face cards)
Among the 13 clubs, 3 are face cards (Jack, Queen, King of clubs).
So, these 3 cards are counted in both previous groups, and we need to subtract them once.
Step 5: Apply inclusion-exclusion principle
P(club or face card) = P(club) + P(face card) − P(club face card)
So, number of favorable outcomes:
= 13 (clubs) + 12 (face cards) − 3 (club face cards)
= 22 favorable outcomes
Step 6: Write probability as a fraction
Probability = favorable outcomes / total outcomes
= 22 / 52
We simplify the fraction:
22 ÷ 2 = 11
52 ÷ 2 = 26
✅ Final Answer: 11/26
Explanation (300 words)
To calculate the probability of drawing a club or a face card from a standard deck of 52 playing cards, we use the rule for combining probabilities when events overlap. A standard deck has four suits—clubs, diamonds, hearts, and spades—and each suit has 13 cards. Among these are face cards, which are the Jack, Queen, and King.
There are 13 club cards and 12 face cards in total (3 in each of the 4 suits). However, 3 of those face cards—the Jack, Queen, and King of clubs—are both clubs and face cards. If we simply added the 13 clubs and 12 face cards, we would double-count those 3 cards. So, we must subtract them once.
That gives us:
13 (clubs) + 12 (face cards) − 3 (face cards that are clubs) = 22 unique favorable cards.
Now, we divide the number of favorable outcomes by the total number of possible outcomes (52 cards) to get the probability: 2252=1126\frac{22}{52} = \frac{11}{26}
This is the simplified probability of drawing either a club or a face card in a single random draw.