deckofcards

Deck Of Cards Lab 2
Probabilities Based on Counting



    Before you can complete this lab, should have some experience with a deck of cards and counting the number of ways an event can occur.

    Read page "Deck of Cards 1" on counting before you begin this page.

    Click on the picture of the playing cards to see what they are called. Then you may answer the questions below.

    Once you complete a question, mouseover the "Answer" button to read the answer.



Probability

      A probability is a fraction which divides the "number of ways an event can happen" by "the total number of events."

More accurately stated, the probability of an event,

 
(frequency of the event)
P(event) =
 
(number of events in the sample space)

"Event" means a result or a thing that happens. "Frequency" means the number of times it happens. "Sample space" means the set of all possible results or events. For more info read "Basic Probability".



State the Probability
  1. If you use just the aces, what's the probability you get a heart?

    1/4, one out of four
    (1 way to get a heart) / (4 possible aces)
     
  2. If you use just the hearts, what's the probability you get a ace?

    1/13, one out of 13
    (1 way to get an ace) / (13 possible hearts)
  3. Considering just the black cards, what's the probability you get a heart? 0/26=0. You can't get a heart using black cards.
  4. If you use just the clubs, what's the probability you get a face card? 3/13
  5. Considering just the aces, what's the probability you get a ace? 4/4=1. You must get an ace.
  6. Given all the cards, what's the probability you get a ace? 4/52 = 1/13. There are 4 aces in the 52 cards.
  7. Given all the cards, what's the probability you get a face card? 12/52 = 3/13


Given A Deck of Cards Find the Probabilities

Given a regular deck of cards, draw 1 card,

  1. What's the probability you get a heart? 13/52 or 1/4
  2. What's the probability you get a club? 13/52 or 1/4
  3. What's the probability you get a ace? 4/52 = 1/13
  4. What's the probability you get a face card? 12/52 = 3/13
  5. What's the probability you get a black card? 26/52 or 1/2
  6. What's the probability you get the King of Clubs? 1/52
  7. What's the probability you get a card that is an ace and at the same time a club? 1/52
  8. What's the probability you get a card that is an ace or is a club? 16/52 = 8/26 = 4/13


Find Probabilities for Cards Drawn Without Replacement

      Without replacement means you take a card, then, without returning it to the deck take a second card.

  1. If you use just the black cards, what's the probability you get a ace then a two?

    (2x2) / (26 x 25) = 4/325

    There are 2 ways to get an ace and two ways to get at 2.
    First there are 26 cards, then there are only 25 cards left.
     
  2. Considering just the black cards, what's the probability you get a face card then another face card?

    (6x5) / (26 x 25) = 3/65
    First there were 6 face cards, then one was drawn so 5 are left.
    First there were 26 cards, then one was drawn so 25 are left.

     
  3. Considering the full deck, what's the probability you get a face card then another face card? (12x11)/(26x25)=33/169, which is (12/26) x (11/25)
  4. If you use just the clubs, what's the probability you get a face card then another face card? (3x2)/(13x12)=1/26, which is (3/13) x (2/12)
  5. Considering the entire deck, what's the probability you get a deuce then a face card? (4x12)/(52x51)=4/221, which is (4/52) x(12/51)
  6. Considering the full deck, what's the probability you get a heart then a second heart? (13x12)/(52x51)=1/17, which is (13/52) x (12/51).


Find the Probability When You Draw Two Cards With Replacement

      With replacement means you take a card, then, return it to the deck, then take a second card.

  1. If you use just the black cards, what's the probability you get a ace then a two? (2x2)/(52x52)=1/676, which is (2/52) x (2/52)
  2. Considering just the black cards, what's the probability you get a face card then another face card? (6x6)/(26x26)=9/169, which is (6/26) x (6/26)
  3. Considering the full deck, what's the probability you get a face card then another face card? (12x12)/(52x52)=9/169, which is (12/52) x (12/52)
  4. If you use just the clubs, what's the probability you get a face card then another face card? (3x3)/(13x13)=9/169, which is (3/13) x (3/13)
  5. Considering the entire deck, what's the probability you get a deuce then a face card? (4x12)/(52x52)=3/169, which is (4/52) x (12/52)
  6. Considering the full deck, what's the probability you get a heart then a second heart? (13x13)/(52x52)=1/16, which is (13/52) x (13/52).


Compute Probabilities
  1. What's the probability you draw 3 face cards when you draw 3 cards from the face cards? (12x11x10)/(12x11x10)=1, which is (12/12) x ( 11/11) x (10/10)
  2. What's the probability you draw 3 face cards when you draw 3 cards without replacement from a deck of cards? (12x11x10)/(52x51x50)=11/1105, which is (12/52) x (11/51) x (10/50)
  3. What's the probability you draw a heart then a club then a diamond when 3 cards are drawn without replacement from the deck of cards? (13x13x13)/(52x51x50) is about .0166, which is (13/52)x(13/51) x(13/50)
  4. What's the probability you draw a heart then a club then a diamond when 3 cards are drawn with replacement from the deck of cards? (13x13x13)/(52x52x52) is about .0156, which is (13/52)x(13/51) x(13/50)
  5. What's the probability you draw the ace of clubs then a heart when you draw 2 cards from the full deck without replacement? (1x13)/(52x51)=1/204, which is (1/52) x (13/51)
  6. What's the probability you draw a 5 then a 4 then an ace from the deck without replacement? (4x4x4)/(52x51x50) is about .00048, which is (4/52) x (4/51) x (4/50)


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© A. Azzolino March 9, 2012
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