Probability with a Deck of Cards

The Structure of a Deck of 52 Cards

A standard deck has 52 cards in four suits:

• Red suits: hearts $\heartsuit$ and diamonds $\diamondsuit$ — 13 cards each (26 red in all).
• Black suits: spades $\spadesuit$ and clubs $\clubsuit$ — 13 cards each (26 black in all).

Each suit has 13 cards: Ace, 2, 3, …, 10, Jack, Queen, King.

Key Point: 52 cards $=$ 4 suits $\times$ 13. Two colours (26 red, 26 black). Learn this layout cold — every card question is counting within it.

Face Cards, Aces and Number Cards

• Face cards are the Jack, Queen and King — $3$ per suit, so $3\times 4 = 12$ face cards in all.
• Aces: one per suit, so $4$ aces.
• Number cards (2 to 10): $9$ per suit, so $36$ in all.
• Court/picture cards in board questions usually means the 12 face cards (some questions include aces — read carefully).

Key Point: 12 face cards (J, Q, K), 4 aces, 4 of each rank, 13 of each suit.

Standard Card Probabilities

For a single card drawn from a well-shuffled deck (total 52):

• $P(\text{a king}) = \dfrac{4}{52} = \dfrac{1}{13}$,
• $P(\text{a red card}) = \dfrac{26}{52} = \dfrac12$,
• $P(\text{a spade}) = \dfrac{13}{52} = \dfrac14$,
• $P(\text{a face card}) = \dfrac{12}{52} = \dfrac{3}{13}$,
• $P(\text{an ace}) = \dfrac{4}{52} = \dfrac{1}{13}$,
• $P(\text{a red king}) = \dfrac{2}{52} = \dfrac{1}{26}$,
• $P(\text{the king of hearts}) = \dfrac{1}{52}$.

Key Point: Denominator is 52; the numerator is how many of that kind exist.

Using the Complement with Cards

"Not" questions are quick with the complement:

• $P(\text{not a king}) = 1 - \dfrac{4}{52} = \dfrac{48}{52} = \dfrac{12}{13}$,
• $P(\text{not a face card}) = 1 - \dfrac{12}{52} = \dfrac{40}{52} = \dfrac{10}{13}$.

If some cards are removed first (e.g. "all kings are removed"), reduce both the count and the total (here to 48) before computing.

Key Point: If cards are removed, the total 52 changes — update the denominator.

Solved Examples

Example 1: A face card

One card is drawn from a well-shuffled deck of 52. Find the probability that it is a face card.

Solution:

1. Face cards $= 12$; total $= 52$.
2. $P = \dfrac{12}{52} = \dfrac{3}{13}$.

Final Answer: $\dfrac{3}{13}$.

Takeaway: 12 face cards (J, Q, K of four suits).

Example 2: A red king

Find the probability of drawing a red king.

Solution:

1. Red kings: king of hearts and king of diamonds $= 2$.
2. $P = \dfrac{2}{52} = \dfrac{1}{26}$.

Final Answer: $\dfrac{1}{26}$.

Takeaway: Two red kings out of 52.

Example 3: Not a king

Find the probability that the card drawn is not a king.

Solution:

1. $P(\text{king}) = \dfrac{4}{52} = \dfrac1{13}$.
2. $P(\text{not king}) = 1 - \dfrac1{13} = \dfrac{12}{13}$.

Final Answer: $\dfrac{12}{13}$.

Takeaway: Complement is quicker than counting all 48 non-kings.

Example 4: Kings removed first

All four kings are removed from a deck; a card is then drawn from the remaining 48. Find the probability that it is a queen.

Solution:

1. Remaining cards $= 48$; queens still $= 4$.
2. $P = \dfrac{4}{48} = \dfrac{1}{12}$.

Final Answer: $\dfrac{1}{12}$.

Takeaway: When cards are removed, use the new total (48).