# Card Probability

In this mini lesson, you will be introduced to the concept of probability of drawing a card from a pack of 52 cards. You will learn interesting facts around deck of cards, suits in a deck of cards, and types of cards in a deck. You will also know more about number of cards in a deck and face cards in a deck by gathering more knowledge about 52 card deck and spades, hearts, diamonds, clubs in a pack of cards.

Here's a trivia fact for you. Did you know that the playing cards that we use today has transformed from when they began, centuries ago. It is said it took about hundred years that involved travel, that made the present day 52 card deck culturally diverse as it was used by travellers back then. It was believed that playing cards originated from China during the Tang Dynasty.

Do you think if you were given the well-shuffled deck of cards shown above, would you draw a queen in the first go?

Well, you may or may not draw a queen in the first go.

Is it possible to say that you might end up drawing a king and instead of a card which is queen from those 52 cards?

Yes, it is possible you might end up drawing a king instead of a queen from those 52 cards as you may or may not draw a queen in the first go.

Stay tuned to find out answers like probability of drawing a king!

**Lesson Plan**

**What Do You Mean by a Deck of Cards?**

Deck of cards is the term used for a set of 52 cards consisting of different types of cards.

It is also commonly referred to as a pack of cards.

- The sample space for a set of cards is 52 as there are 52 cards in a deck. This makes the denominator for finding the probability of drawing a card as 52.
- Learn more about related terminology of probability to solve problems on card probability better.

**What Are the Types of Cards in a Deck?**

We can classify types of cards in a deck in two ways:

- Based on color of cards
- Based on suits

**Baesd on Colour of Cards**

There are two colors of cards in each deck:

- Red
- Black

The suits which are represented by red cards are hearts and diamonds while the suits represented by black cards are spades and clubs.

There are 26 red cards and 26 black cards.

Let's learn about the suits in a deck of cards.

**Based on Suits**

Suits in a deck of cards are the representations of red and black color on the cards.

Based on suits, the types of cards in a deck are:

- Spades
- Hearts
- Diamonds
- Clubs

Let's see what each suit represents:

- Spades

- Hearts

- Diamonds

- Clubs

There are 52 cards in a deck.

Each card can be categorized into 4 suits constituting 13 cards each.

There is one more categorization of a deck of cards:

- Face cards
- Number cards
- Aces

**Face Cards**

These cards are also known as court cards.

They are Kings, Queens, and Jacks in all 4 suits.

** **

**Number Cards**

All the cards from 2 to 10 in any suit are called the number cards.

These cards have numbers on them along with each suit being equal to the number on number cards.

**Aces**

There are 4 Aces in every deck, 1 of every suit.

- There are 13 cards of each suit, consisting of 1 Ace, 3 face cards, and 9 number cards.
- There are 4 Aces, 12 face cards, and 36 number cards in a 52 card deck.
- Probability of drawing any card will always lie between 0 and 1.
- The number of spades, hearts, diamonds, and clubs is same in every pack of 52 cards.

Now that you know all about facts about a deck of cards, you can draw a card from a deck and find its probability easily.

**How to Determine the Probability of Drawing a Card?**

Let's learn how to find probability first.

Now you know that probability is the ratio of number of favorable outcomes to the number of total outcomes, let's apply it here.

**Examples**

**Example 1: **What is the probability of drawing a king from a deck of cards?

Solution: Here the event E is drawing a king from a deck of cards.

There are 52 cards in a deck of cards.

Hence, total number of outcomes = 52

The number of favorable outcomes = 4 (as there are 4 kings in a deck)

Hence, the probability of this event occuring is

P(E) = 4/52 = 1/13

\(\therefore\) Probability of drawing a king from a deck of cards is 1/13. |

**Example 2:** What is the probability of drawing a black card from a pack of cards?

Solution: Here the event E is drawing a black card from a pack of cards.

The total number of outcomes = 52

The number of favorable outcomes = 26

Hence, the probability of event occuring is

P(E) = 26/52 = 1/2

\(\therefore\) Probability of drawing a black card from a pack of cards is 1/2. |

**Solved Examples**

Example 1 |

Jessica has drawn a card from a well-shuffled deck. Help her find the probability of the card either being red or a King.

**Solution**

Jessica knows here that event E is the card drawn being either red or a King.

The total number of outcomes = 52

There are 26 red cards, and 4 cards which are Kings.

However, 2 of the red cards are Kings.

If we add 26 and 4, we will be counting these two cards twice.

Thus, the correct number of outcomes which are favorable to E is

26 + 4 - 2 = 28

Hence, the probability of event occuring is

P(E) = 28/52 = 7/13

\(\therefore\) Probability of card either being red or a King card is 7/13. |

Example 2 |

Help Diane determine the probability of the following:

- Drawing a Red Queen
- Drawing a King of Spades
- Drawing a Red Number Card

**Solution**

Diane knows here the events E_{1}, E_{2}, and E_{3} are Drawing a Red Queen, Drawing a King of Spades, and Drawing a Red Number Card.

The total number of outcomes in every case = 52

- Drawing a Red Queen

There are 26 red cards, of which 2 are Queens.

Hence, the probability of event E_{1 }occuring is

P(E_{1}) = 2/52 = 1/26

- Drawing a King of Spades

There are 13 cards in each suit, of which 1 is King.

Hence, the probability of event E_{2 }occuring is

P(E_{2}) = 1/52

- Drawing a Red Number Card

There are 9 number cards in each suit and there are 2 suits which are red in color.

There are 18 red number cards.

Hence, the probability of event E_{3 }occuring is

P(E_{3}) = 18/52 = 9/26

\(\therefore\) Diane determined that the probabilities are P(E_{1}) = 1/26, P(E_{2}) = 1/52, and P(E_{3}) = 9/26. |

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about probability of drawing a card from a pack of 52 cards with the practice questions. Now you will easily be able to solve problems on number of cards in a deck, face cards in a deck, 52 card deck, spades hearts diamonds clubs in pack of cards. Now you can draw a card from a deck and find its probability easily .

The mini-lesson targeted the fascinating concept of card probability. The math journey around card probability starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that is not only relatable and easy to grasp, but will also stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students! Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic. Be it problems, online classes, videos, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**FAQs on Card Probability**

### 1. How do you calculate the probability of drawing a card?

We find the ratio of the favorable outcomes as per the condition of drawing the card to the total number of outcomes, i.e, 52.

### 2. What is the probability of drawing any face card?

Probability of drawing any face card is 6/26.

### 3. What is the probability of drawing a red card?

Probability of drawing a red card is 1/2.

### 4. What is the probability of drawing a king or a red card?

Probability of drawing a king or a red card is 7/13.

### 5. What is the probability of drawing a king or a queen?

The probability of drawing a king or a queen is 2/13.

### 6. What are the 5 rules of probability?

The 5 rules of probability are:

- Rule 1

For any event E, the probability of occurence of E will always lie between 0 and 1

- Rule 2

The sum of probabilities of every possible outcome will always be 1

- Rule 3

The sum of probability of occurence of E and probability of E not occuring will always be 1

- Rule 4

When any two events are not disjoint, the probability of occurence of A and B is not 0 while when two events are disjoint, the probability of occurence of A and B is 0.

- Rule 5

As per this rule, P(A or B) = (P(A) + P(B) - P(A and B)).

### 7. What is the probability of drawing a king of hearts?

Probability of drawing a king of hearts is 1/52.

### 8. Is Ace a face card in probability?

No, Ace is not a face card in probability.

### 9. What is the probability it is not a face card?

The probability it is not a face card is 10/13.

### 10. How many black non-face cards are there in a deck?

There are 20 black non-face cards in a deck.

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- Practice worksheets in and after class for conceptual clarity
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