Card probability is the probability of the events involving a deck of playing cards. In simple words, card probability is one part of probability in which we find the probability of drawing a card from the deck of cards.
- Deck of Cards is a collection of 52 cards.
- Divided into four suits namely Spade (♠), Club (♣), Heart (
\color{red}{\text{♥}} ), and Diamond (\color{red}{\blacklozenge} ). - For a single chosen card, the sample space is 52 i.e., the total number of outcomes for a single chosen card from a deck is 52.
Types
Any deck of cards can be classified in many ways, some of the parameters on which cards can be classified are:
1. Based on Colors
Based on colors a deck of cards can be classified into two categories,
- Red Cards
- Black Cards
A total of 52 cards are divided equally into red and black cards which means there are 26 red cards and 26 black cards in the deck.
2. Based on Suits
Four suits in the deck of cards are:
- Hearts (
\color{red}{\text{♥}} ) - Diamonds (
\color{red}{\blacklozenge} ) - Clubs (♣)
- Spades (♠)
Other than these there is one more classification of cards, based on the rank of cards:
1. Ace
Ace is one such card which either is the most important or least important based on the game. This card "A" written on it and each suit has one of such card i.e., four ace cards.
2. Number Cards
From 2 to 10, there are 9 cards per suit, thus there is a total of 36 such cards.
3. Face Cards
Face cards as the name suggests, contain a figure or face of the figure on the card. There are three cards of each suit i.e., Jack, Queen, King. Thus there are a total of 12 face cards.
All these classifications can be seen in the following table.
| Deck Of Cards (52 cards) | ||||
|---|---|---|---|---|
| Colored Cards | Black Cards (26 cards) | Red Cards (26 cards) | ||
| Suits | Spade (13 cards) | Club (13 cards) | Heart (13 cards) | Diamond (13 cards) |
| Face Cards (12 cards in a deck and 3 cards in each suit) | K (King) | K (King) | K (King) | K (King) |
| Q (Queen) | Q (Queen) | Q (Queen) | Q (Queen) | |
| J (Jack) | J (Jack) | J (Jack) | J (Jack) | |
| Number Cards (36 cards in a deck and 9 cards in a suit) | 10 | 10 | 10 | 10 |
| 9 | 9 | 9 | 9 | |
| 8 | 8 | 8 | 8 | |
| 7 | 7 | 7 | 7 | |
| 6 | 6 | 6 | 6 | |
| 5 | 5 | 5 | 5 | |
| 4 | 4 | 4 | 4 | |
| 3 | 3 | 3 | 3 | |
| 2 | 2 | 2 | 2 | |
| Ace Cards (4 cards in a deck and 1 card in a suit) | A (Ace) | A (Ace) | A (Ace) | A (Ace) |
Playing Card Probability
Some of the common events in card probabilities are discussed in the following table:
Event E for drawing card | Probability P(E) |
|---|---|
An Ace | P(E) = 4 / 52 = 1 / 13 |
A King | P(E) = 4 / 52 = 1 / 13 |
A Number Card | P(E) = 36 / 52 = 9 / 13 |
A Face Card | P(E) = 12 / 52 = 3 / 13 |
A Spade Card | P(E) = 13 / 52 = 1 / 4 |
A Red Card | P(E) = 26 / 52 = 1 / 2 |
Steps to Calculate
The steps to find the probability of events involving cards are the same as all the other probabilities, which are given as follows:
Step 1: First, find the number of favorable outcomes from the given question.
Step 2: Then, find the total number of outcomes.
Step 3: Apply the probability formula to find the card probability.
Example: What is the probability of drawing an ace from a deck of cards.
Answer:
Here, E is event of drawing an ace card
Total number of outcomes in a deck n(S) = 52
Number of favorable outcomes = n(E) = drawing an ace card from deck = 4 (There are 4 ace cards in 1 deck)
P(E) = n(E) / n(S) = 4 / 52
P(E) = 1 / 13
Probability of drawing an ace card = 1 / 13
Probability of Drawing Two Cards
When two cards are drawn from a deck, the probability depends on whether the first card is returned to the deck or not.
1. Probability Without Replacement : In this case, the first card is not returned to the deck before drawing the second card. Therefore, the total number of cards decreases from 52 to 51.
Example: Find the probability of drawing two Aces consecutively without replacement.
Solution:
Probability of drawing an Ace on the first draw:
P(A_1)=\frac{4}{52} After drawing one Ace, 3 Aces remain among 51 cards.
Probability of drawing an Ace on the second draw:
P(A_2\mid A_1)=\frac{3}{5} Using the multiplication rule:
P(A_1\cap A_2)=\frac{4}{52}\times\frac{3}{51}
=\frac{12}{2652}
{P(A_1\cap A_2)=\frac{1}{221}}
2. Probability With Replacement : In this case, the first card is returned to the deck before drawing the second card. Therefore, the total number of cards remains 52 for both draws.
Example: Find the probability of drawing an Ace twice with replacement.
Solution:
Probability of drawing an Ace on the first draw:
P(A_1)=\frac{4}{52} Since the first card is replaced, the probability of drawing an Ace again is:
P(A_2)=\frac{4}{52} Using the multiplication rule:
P(A_1\cap A_2)=\frac{4}{52}\times\frac{4}{52} = 16 / 2704
{P(A_1\cap A_2)=\frac{1}{169}}
Applications
Card probability is useful for understanding how probability works in games and situations involving random selection.
- Card Games: It helps calculate the chances of getting a particular card or combination of cards in games such as Poker and Rummy.
- Understanding Random Events: Drawing a card provides a simple way to understand sample space, favorable outcomes, and probability.
- Decision-Making in Games: Players can use probability to understand the likelihood of obtaining a required card.
- Learning Probability Rules: Card-based examples help explain the addition rule, multiplication rule, complementary probability, and conditional probability.
