3 cards are picked at random from a pack of 52 cards, what is the probability of getting 3 queens

Probability is a field of mathematics that studies the likelihood of a random event occurring. Since many events cannot be predicted with total certainty, we use probability to anticipate how probable they are to occur. Probability can range from 0 to 1, with 0 indicating an improbable event and 1 indicating a certain event. Probability has many applications. Risk assessment and modeling are examples of how probability theory is used in everyday life. Actuarial science is used by the insurance sector and markets to establish pricing and make trading decisions. Environmental control, entitlement analysis, and financial regulation all use probability methodologies. Probability also finds its applications in weather forecasting, agriculture, and politics.

Formula for Probability

Probability of an event, P(A) = (Number of favorable outcomes) / (Total number of outcomes)  

There are majorly three types of probability, they are theoretical probability, experimental probability, and axiomatic probability. Let’s learn about them in detail,

Theoretical Probability

It is predicated on the likelihood of something occurring. The rationale behind probability is the foundation of theoretical probability. For example, to calculate the theoretical probability of rolling a die and getting the number 3, we must first know the number of possible outcomes. We know that a die has six numbers (i.e. 1, 2, 3, 4, 5, 6), hence the number of possible outcomes is six as well. So, the probability of rolling a three on a dice is one in six, or 1/6

Experimental Probability

Experimental probability, unlike theoretical probability, incorporates the number of trials, i.e. it is based on the results of an experiment. The experimental probability can be computed by dividing the total number of trials by the number of possible outcomes. For example, if a dice is rolled 40 times and the number three is recorded 10 times then, the experimental probability for heads is 10/40 or 1/4

Axiomatic Probability

A set of principles or axioms are established in the axiomatic probability that applies to all types. Kolmogorov established these axioms, which are known as Kolmogorov’s three axioms. There are three main concepts in probability. They are sample space, events, and probability function. Let’s learn about them in detail,

Sample Space (S)

A sample space is the collection of all possible outcomes of an experiment. Tossing three dice produces a sample space of 216 potential outcomes, each of which may be recognized by an ordered set (a, b, c), where a, b, and c take one of the following values: 1, 2, 3, 4, 5, 6.

Event (A)

A well-defined subset of the sample space is referred to as an event. The event the sum of the faces shown on the two dice equals five has six outcomes: (1, 1, 3), (1, 3, 1), (3, 1, 1), (1, 2, 2), (2, 2, 1) and (2, 1, 1).

Probability Function (P)

The function that is used to assign a probability to an occurrence is known as the Probability Function (P). The probability function (P) determines the likelihood of an event (A) being drawn from the sample space (S).

Solution:

Total number of cards in a deck = 52

Total number of kings in a deck of 52 cards = 4

If we pick one card at random from the 52 cards, the probability of getting a king = Total number of kings in the deck / Total number of cards in the deck.

i.e. Probability of getting a king = 4/52 = 1/13

Total number of queens in a deck of 52 cards = 4

If we pick one card at random from the 52 cards, the probability of getting a queen = Total number of queens in the deck / Total number of cards in the deck.

i.e. Probability of getting a queen = 4/52= 1/13

Therefore, probability of getting a king or a queen, P(E) = probability of getting a king + probability of getting a queen = 1/13 + 1/13 = 2/13

Similar Questions

Question 1:  Find the probability of getting a red king.

Solution:

Total number of cards = 52

No. of favorable cards that are red kings = 2

Therefore probability of getting a red king = 2/52 = 1/26

Question 2: Find the probability of getting a red non-face card.

Solution:

Total number of red cards in a deck = 26

Face cards are cards that are either, king, queen, or jack

Total number of red face cards = 6

Therefore, the total number of red non-face cards = 26 – 6

Therefore, the probability of getting a red non-face card= 20/52 = 5/13

Question 3: Find the probability of getting a black card.

Solution:

Total number of cards = 52

Total number of suits = 4

Total number of black suits = 2

Therefore, total number of black cards = 2 × 13 = 26

Therefore the probability of getting a black card= Total number of black cards in the deck / total number of cards in the deck = 26/52 = 1/2.

Question 4: Find the probability of getting a red ace or a spade.

Solution:

Total number of cards = 52

No. of favorable cards that are red aces = 2

Therefore the probability of getting a red ace = 2/52

Total number of cards that are spades =13

Therefore the probability of getting a spade = 13/52

Therefore, probability of getting a red ace or a spade, P(E) = probability of getting a red ace + probability of getting a spade = 2/52 + 13/52 = 15/52

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Important Notes

• 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. 

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.

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: 

There are 52 cards in a deck.

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

These cards are also known as court cards.

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

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. 

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

Tips and Tricks

• 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

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

Solved Examples

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

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 E1, E2, and E3 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

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

Hence, the probability of event E1 occuring is

P(E1) = 2/52 = 1/26

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

Hence, the probability of event E2 occuring is

P(E2) = 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 E3 occuring is

P(E3) = 18/52 = 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.

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.

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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:

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

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

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

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.

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.