Probability and Playing Cards

In maths, the probability is nothing but the chance of occurrence of an event successfully. Examples of probability include tossing a coin, rolling a die, playing cards and so on. This probability and playing cards page contains the example questions and how to calculate the probability of playing cards.

Basics About Playing Cards Probability

In a pack of playing cards, we can see 52 cards which are divided into 4 suits having each 13 cards. The shape names of the suits are spades, diamonds, hearts and clubs. Again these suits are available in two different colours like red and black. The color of diamonds and hearts in red and spades, clubs color is black.

Each suit have Ace, King, Queen, Jack, 10, 9, 8, 7, 6, 5, 4, 3, 2. Based on the definition, the formula to calculate the probability is here:

Probability = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \) (or)

P(E) = \(\frac { N(E) }{ N(S) } \)

For playing cards, N(S) is always 52.

Also, Check

How to Calculate Probability and Playing Cards?

Below mentioned are the simple steps to find the probability of playing cards easily.

  • Find the number of favorable events.
  • We already know that a total number of possible outcomes for playing cards is 52.
  • So, divide the number of favorable events by 52 to get the answer.

Worked out Problems on Probability and Playing Cards

Problem 1:
A card is drawn from a pack of 52 cards. What is the probability that the drawn card is king?

Solution:
Let E be the event of drawing a king card.
There are 4 king cards in playing cards.
Number of favorable outcomes N(E) = 4
Total number of outcomes N(S) = 52
Probability of drawing a king from pack of cards = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 4 }{ 52 } \) = \(\frac { 1 }{ 13 } \)

Problem 2:
A card is drawn at random from a well-shuffled pack of cards numbered 1 to 20. Find the probability of
(i) getting a number less than 5
(ii) getting a number divisible by 3

Solution:
Total number of possible outcomes N(S) = 20
(i) Number of favorable outcomes = Number of cards showing less than 5 = 4 i.e {1, 2, 3, 4}
Probability of getting a number less than 5 P(E) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 4 }{ 20 } \) = \(\frac { 1 }{ 5 } \)
(ii) Number of favorable outcomes = Number of cards showing a number divisible by 3 = 6 i.e {3, 6, 9, 12, 15, 18}
Probability of getting a number divisible by 3 P(E) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 6 }{ 20 } \) = \(\frac { 3 }{ 10 } \)

Problem 3:
All kings, jacks, diamonds have been removed from a pack of 52 playing cards and the remaining cards are well shuffled. A card is drawn from the remaining pack. Find the probability that the card drawn is:
(i) a red queen
(ii) a face card
(iii) a black card
(iv) a heart

Solution:
Number of kings in a pack of 52 cards = 4
Number of jacks in a pack of 52 cards = 4
Number of diamonds in a pack of 52 cards = 13
Total number of removed cards = (4 + 4 + 11) = 19 cards
[Excluding the diamond king and jack there are 11 diamonds]
Remaining total cards after removing kings, jacks and diamonds = 52 – 19 = 33
(i)
Queen of heart and queen of diamond are two red queens
The queen of diamond is already removed.
So, there is 1 red queen out of 33 cards
Therefore, the probability of getting ‘a red queen’ P(A) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 1 }{ 33 } \)
(ii)
Number of face cards after removing all kings, jacks, diamonds = 3
Therefore, the probability of getting ‘a face card’ P(B) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 3 }{ 33 } \) = \(\frac { 1 }{ 11 } \)
(iii)
Cards of spades and clubs are black cards.
Number of spades = 13 – 2 = 11, since king and jack are removed
Number of clubs = 13 – 2 = 11, since king and jack are removed
Therefore, in this case, total number of black cards = 11 + 11 = 22
The probability of getting ‘a black card’ P(C) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 22 }{ 33 } \) = \(\frac { 2 }{ 3 } \)
(iv)
Number of hearts = 13
Therefore, in this case, total number of hearts = 13 – 2 = 11, since king and jack are removed
Therefore, the probability of getting ‘a heart card’ P(D) = \(\frac { Number of favorable outcomes }{ Total number of outcomes } \)
= \(\frac { 11 }{ 33 } \) = \(\frac { 1 }{ 3 } \).

Frequently Asked Question’s

1. How are 52 cards divided?

A standard pack of playing cards have 52 cards. All those cards are divided into two colors red & black. Deck of cards contains 4 suits “spades”, “hearts”, “clubs”, “diamonds”. Each suit has 13 cards Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King. Hearts & diamonds are in red color and spades, clubs are in black color.

2. What is the probability of getting a number card from a deck of 52 cards?

Total number of cards = 52

Number of number cards in pack = 36

Probability of getting a number card = \(\frac { 36 }{ 52 } \) = \(\frac { 9 }{ 13 } \)

3. What are the 4 types of cards?

The 4 different types of cards are clubs, diamonds, hearts and spades.

4. How many kings are in a pack of cards?

A total of 4 kings are there in a pack of cards.

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