About Playing Cards Puzzles
A card puzzle asks for the probability that a description is true of the card or cards drawn, so the work is to count how many cards, or how many pairs, satisfy the description and divide by the number of ways the draw can happen. The counting is the whole question, and the usual errors come from counting a card twice.
What you need to understand
- A standard pack has 52 cards: thirteen ranks in each of four suits.
- Hearts and diamonds are red, spades and clubs black, so there are 26 cards of each colour.
- There are twelve picture cards, three in each suit.
- For a single card the probability is the number of matching cards divided by 52.
- For two cards the order does not matter, so there are 52 x 51 / 2 = 1326 possible pairs.
- For an or description, add the two counts and subtract the cards belonging to both.
- For a neither description, count the cards that do satisfy it and subtract from 52.
How to work through these questions
- Read the description and decide whether it names one group or a union of two.
- Count the cards that match. For a union, add the two counts and subtract the overlap.
- For a neither description, count the matching cards and subtract from 52.
- Divide by 52 and reduce the fraction.
- For two cards, count pairs rather than cards, remembering that order does not matter.
- Check the result is sensible: a description covering a quarter of the pack should give about one in four.
Mistakes that cost marks
- Double counting the cards that fall in both parts of an or description.
- Treating the two draws as ordered, which doubles the favourable and total counts together and hides the error if only one is doubled.
- Counting aces as picture cards, when picture cards are only the jack, queen and king.
- Mixing up the 13 cards of a suit with the 26 cards of a colour.
- Leaving the fraction unreduced when the options are reduced.
- Counting the cards that do not match when the question asks for those that do.
Worked example
A card is drawn at random from a well-shuffled pack of 52 playing cards. What is the probability that it is a king or a heart?
- There are 4 kings in the pack.
- There are 13 hearts in the pack.
- The king of hearts belongs to both groups, so adding 4 and 13 counts it twice. The number of matching cards is 4 + 13 - 1 = 16.
- The probability is 16/52, which reduces to 4/13.
Answer: 4/13
Practice questions with answers
A few Playing Cards Puzzles questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
A card is drawn at random from a standard pack of 52 playing cards. What is the probability that it is a spade?
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A
1/2
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B
17/52
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C
3/4
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D
1/4
Answer: Option D — with explanation
Of the 52 cards, 13 are a spade, so the probability is 13/52, which reduces to 1/4.
Common mistakes
- assumed the red and black cards alone: 1/2
- counted four extra cards as matching: 17/52
- counted the cards that do not match: 3/4
Question 2
Two cards are drawn at random from a well-shuffled pack of 52 playing cards. What is the probability that both are of different suits?
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A
10/13
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B
13/17
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C
4/17
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D
13/34
Answer: Option B — with explanation
Two cards can be drawn from a pack in 52 x 51 / 2 = 1326 ways, because the order the two cards arrive in does not matter.
Of those, 1014 give a result where both are of different suits, so the probability is 1014/1326 = 13/17.
Common mistakes
- allowed for a few extra pairs: 13/34
- counted the two cards as ordered, doubling the favourable cases: 4/17
- counted the pairs that do not match: 10/13
Question 3
A card is drawn at random from an ordinary deck of 52 cards. What is the chance that it is a red card?
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A
7/13
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B
1/4
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C
1/2
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D
15/26
Answer: Option C — with explanation
Of the 52 cards, 26 are a red card, so the probability is 26/52, which reduces to 1/2.
Common mistakes
- counted four extra cards as matching: 7/13
- assumed the red and black cards alone: 15/26
- assumed one suit out of four: 1/4
Question 4
Two cards are drawn together at random from a standard pack of 52 playing cards. What is the chance that both are picture cards?
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A
22/221
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B
11/221
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C
12/221
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D
210/221
Answer: Option B — with explanation
Two cards can be drawn from a pack in 52 x 51 / 2 = 1326 ways, because the order the two cards arrive in does not matter.
Of those, 66 give a result where both are picture cards, so the probability is 66/1326 = 11/221.
Common mistakes
- allowed for a few extra pairs: 12/221
- counted the two cards as ordered, doubling the favourable cases: 22/221
- counted the pairs that do not match: 210/221
Question 5
A card is drawn at random from a well-shuffled pack of 52 playing cards. What is the chance that it is a king or a heart?
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A
4/13
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B
1/4
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C
9/13
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D
1/2
Answer: Option A — with explanation
Of the 52 cards, 16 are a king or a heart, so the probability is 16/52, which reduces to 4/13.
Common mistakes
- assumed one suit out of four: 1/4
- counted the cards that do not match: 9/13
- assumed the red and black cards alone: 1/2
Frequently asked questions
Why is the total 1326 when two cards are drawn?
Because two cards drawn together can be chosen in 52 x 51 / 2 ways. Dividing by two is necessary because the order the cards arrive in does not matter.
How do I handle a description containing or?
Add the counts of the two parts and subtract the cards that satisfy both, which have otherwise been counted twice.
How do I handle a description containing neither?
Count the cards that do satisfy the description, subtract that from 52, and use the result as the numerator.
Are aces picture cards?
No. The picture cards are the jack, the queen and the king. An ace is a rank of its own and is counted separately.
Does it matter which suit is named?
Not on its own. Every suit has thirteen cards, so naming hearts or spades gives the same count unless colour is also mentioned.
