Drawing a card from a pack of 52
A pack has 52 cards in 4 suits of 13. Two suits are red, two black. Kings, queens and jacks are the 12 face cards.
What is in a pack of 52 cards
| Group | How many | P(drawing one) |
|---|---|---|
| Any one suit, e.g. hearts | 13 | 13/52 = 1/4 |
| Red cards (or black cards) | 26 | 26/52 = 1/2 |
| Face cards (K, Q, J) | 12 | 12/52 = 3/13 |
| Kings (or queens, jacks, aces) | 4 | 4/52 = 1/13 |
| Red face cards | 6 | 6/52 = 3/26 |
| One particular card, e.g. the queen of spades | 1 | 1/52 |
Worked example 1: one card, three events
One card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is (i) a black king, (ii) not a face card, (iii) a diamond.
- Formula: P(E) = favourable outcomes ÷ total outcomes. Total = 52.
- Black kings: king of spades, king of clubs (2). P = 2/52 = 1/26.
- Face cards = 12, so cards that are not face cards = 52 − 12 = 40. P = 40/52 = 10/13.
- Diamonds = 13. P = 13/52 = 1/4.
(i) 1/26, (ii) 10/13, (iii) 1/4.
Worked example 2: cards removed from the pack
All four kings are removed from a pack of 52 cards. One card is drawn from the remaining cards. Find the probability that it is (i) a queen, (ii) a red card, (iii) a face card.
- Cards left = 52 − 4 = 48. So total outcomes = 48.
- Formula: P(E) = favourable outcomes ÷ 48.
- All 4 queens are still there. P(queen) = 4/48 = 1/12.
- Red cards left: 26 − 2 red kings = 24. P(red) = 24/48 = 1/2.
- Face cards left: 12 − 4 kings = 8. P(face card) = 8/48 = 1/6.
(i) 1/12, (ii) 1/2, (iii) 1/6.
Face card or not?
Read 'red face card' carefully: only hearts and diamonds count, so there are 6.
Card questions: a checklist
- Write the total first: 52, or fewer if some cards are removed.
- Count each group from the table: a suit has 13, a colour 26, face cards 12, each rank 4.
- If two groups overlap, like hearts and kings, count the shared card only once.
- For 'not' questions, subtract from the total or use 1 − P(E).
- Simplify every fraction fully.
Many families play cards during holidays, and teachers use a pack to explain chance. A well-shuffled pack gives every one of the 52 cards the same chance of being drawn.
1 marks: “A card is drawn from a well-shuffled pack of 52 cards. Find the probability that it is a red queen.” There are 2 red queens (hearts and diamonds) in 52 cards. P = 2/52 = 1/26.
2 marks: “A card is drawn from a pack of 52 cards. Find the probability that it is neither a king nor a queen.” Kings and queens together = 8, so the others = 44. P = 44/52 = 11/13.
3 marks: “The jack, queen and king of clubs are removed from a pack of 52 cards. A card is drawn from the rest. Find the probability that it is (i) a club, (ii) a face card, (iii) a black card.” Total = 52 − 3 = 49. (i) Clubs left = 10, so 10/49. (ii) Face cards left = 9, so 9/49. (iii) Black cards left = 23, so 23/49.
52 = 4 × 13. Red 26, black 26, faces 12 (K, Q, J), each rank 4. The ace has no face.