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Find the probability that if one card drawn from a well-shuffled pack of 52 playing cards is a face card?

  • Last Updated : 21 Sep, 2021
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Probability means Possibility. It states how likely an event is about to happen. The probability of an event can exist only between 0 and 1 where 0 indicates that event is not going to happen i.e. Impossibility and 1 indicates that it is going to happen for sure i.e. Certainty. 
The higher or lesser the probability of an event, the more likely it is that the event will occur or not respectively. 

For example – An unbiased coin is tossed once. So the total number of outcomes can be 2 only i.e. either “heads” or “tails”. The probability of both outcomes is equal i.e. 50% or 1/2.

So, the probability of an event is Favorable outcomes/Total number of outcomes. It is denoted with the parenthesis i.e. P(Event).

P(Event) = (Number of Favorable Outcomes) / (Number of Total Outcomes)

Note: If the probability of occurring of an event A is 1/3 then the probability of not occurring of event A is 1 – P(A) i.e. 1- (1/3) = 2/3



What is a Sample Space?

All the possible outcomes of an event are called Sample spaces.

Examples-

  • A six faced dice is rolled once. So, total outcomes can be 6 and 
    Sample space will be [1, 2, 3, 4, 5, 6]
     
  • An unbiased coin is tossed, So, total outcomes can be 2 and 
    Sample space will be [Head, Tail]
     
  • If two die are rolled together then total outcomes will be 36 and 
    Sample space will be 
    [ (1, 1) (1, 2) (1, 3) (1, 4) (1, 5) (1, 6)  
      (2, 1) (2, 2) (2, 3) (2, 4) (2, 5) (2, 6)  
      (3, 1) (3, 2) (3, 3) (3, 4) (3, 5) (3, 6)  
      (4, 1) (4, 2) (4, 3) (4, 4) (4, 5) (4, 6)  
      (5, 1) (5, 2) (5, 3) (5, 4) (5, 5) (5, 6)  
      (6, 1) (6, 2) (6, 3) (6, 4) (6, 5) (6, 6) ]

Types of Events

Independent Events

If two events (A and B) are independent then their probability will be P(A and B) = P (A ∩ B) = P(A).P(B) i.e. P(A) × P(B)

Example: If two coins are flipped, then the chance of both being tails is 1/2 × 1/2  = 1/4

Mutually exclusive events

  • If event A and event B can’t occur simultaneously, then they are called mutually exclusive events.
  • If two events are mutually exclusive, then the probability of both occurring is denoted as P (A ∩ B) 
    and P (A and B) = P (A ∩ B) = 0
  • If two events are mutually exclusive, then the probability of either occurring is denoted as P (A ∪ B) 
    P (A or B) = P (A ∪ B)                     
                   = P (A) + P (B) − P (A ∩ B)                     
                   = P (A) + P (B) − 0                     
                   = P (A) + P (B)

Example: The chance of rolling a 2 or 3 on a six-faced die is P (2 or 3) = P (2) + P (3) = 1/6 + 1/6 = 1/3



Not Mutually exclusive events

If the events are not mutually exclusive then

P (A or B) = P (A ∪ B) = P (A) + P (B) − P (A and B)

What is a Conditional Probability?

For the probability of some event A, the occurrence of some other event B is given. It is written as P (A ∣ B)

P (A ∣ B) = P (A ∩ B) / P (B)

Example: In a bag of 3 black balls and 2 yellow balls (5 balls in total), the probability of taking a black ball is 3/5, and to take a second ball, the probability of it being either a black ball or a yellow ball depends on the previously taken out ball. Since, if a black ball was taken, then the probability of picking a black ball again would be 1/4, since only 2 black and 2 yellow balls would have been remaining, if a yellow ball was taken previously, the probability of taking a black ball will be 3/4.

Some points related to Cards:

  • There are 52 cards in a deck.
  • In 52 cards, there are 26 cards of each color i.e. 26 red and 26 black cards.
  • In 26 red cards, there are 2 suits of 13 cards each i.e. 13 heart and 13 diamond cards.
  • In 26 black cards, there are 2 suits of 13 cards each i.e. 13 spades and 13 club cards.
  • Each suite has 13 cards from 2 to 10, J, Q, K, and A which means 4 cards of each type.
  • J, Q, and K are known as Face cards.

Find the probability that if one card drawn from a well shuffled pack of 52 playing cards is a face card? 

Solution:

Total number of cards are 52 and number of face cards in 52 cards are 12.



So, total outcomes = 52 
favorable outcomes = 12 (4 Jacks, 4 Queens, 4 Kings)

So, the probability of getting a face card = Favorable outcomes / Total outcomes                                                                                                                        = 12 / 52 = 3/13

P(F) = 3/13

Similar Questions

Question 1: When a single card is drawn from a well-shuffled 52 card deck, then find the probability of getting a non-face card?

Solution:

Total number of cards are 52 and number of non-face cards in 52 cards are 40.

So, total outcomes = 52
favorable outcomes = 40

So, the probability of getting a non-face card = Favorable outcomes/Total outcomes
                                                                       = 40/52 = 10/13
P(Non-Face) = 10/13

Question 2: When a single card is drawn from a well-shuffled 52 card deck, find the probability of getting a red face card?

Solution:



Total number of cards are 52 and number of red face cards in 52 cards are 6.

So, total outcomes = 52
favorable outcomes = 6 (2 Red J, 2 Red Q, 2 Red K)

So, the probability of getting a red face card = Favorable outcomes/Total outcomes
                                                                      = 6/52 = 3/26

P(RF) = 3/26

Question 3: When a single card is drawn from a well-shuffled 52 card deck, find the probability of getting a black face card?

Solution:

Total number of cards are 52 and number of black face card in 52 cards are 6.

So, total outcomes = 52
favorable outcomes = 6 (2 Black J, 2 Black Q, 2 Black K)

So, the probability of getting a black face card = Favorable outcomes/Total outcomes
                                                                         = 6 / 52 = 3/26

P(BF) = 3/26

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