Question

Five cards are dealt with. What is the probability that the 4th one is a king? My answer was 4/52. Then my instructor gave the following comments: I agree with your answer for the "with replacement" scenario. I do however interpret the question where someone is dealing out cards to be a scenario without replacement. Could you also include the without replacement probability? What I meant was an enumeration of the probability that the 4th card dealt is a king, with the various possibilities that a king or more has been dealt in the first three cards. So for example, No king dealt in first 3 cards, followed by king in 4th card: (48/52)*(47/51)*(46/50)*(4/49) Exactly 1 king dealt in first 3 cards, followed by king in 4th card (4/52)*(48/51)*(47/50)*(3/49) * 3, where the 3 represents the 3 ways a sole king can be drawn in the first three cards Then logic would follow for exactly 2 kings in the first 3 (followed by king in 4th card), and then all kings in the first 3 (followed by king in the 4th card). Then logic would follow for exactly 2 kings in the first 3 (followed by king in 4th card), and then all kings in the first 3 (followed by king in the 4th card). Kindly help me understand this. Thanks!

Answer 1

From a deck of 52 cards, 4 cards can be drawn in ways.
In a deck, there are 4 King cards and 48 non-King cards.
All possible combination of the 4th card to be a King card are:

• Scenario 1: Only the fourth card is King
  The first 3 cards can be chosen from 48 cards in ways.
  The fourth card can be chosen from 4 King cards in ways.
  Hence, all possible combinations are: ways.
  
  
• Scenario 2: The fourth card is the 2nd King
  The first King card and 2 non-King cards can be chosen in ways.
  The fourth card can be chosen from the remaining 3 King cards in ways.
  Hence, all possible combinations are: ways.
  
  
• Scenario 3: The fourth card is the 3rd King
  The first two King card and 1 non-King card can be chosen in ways.
  The fourth card can be chosen from the remaining 2 King cards in ways.
  Hence, all possible combinations are: ways.
  
  
• Scenario 4: The fourth card is the 4th King
  The first three King cards and 0 non-King card can be chosen in ways.
  The fourth card can be chosen from the remaining 1 King card in ways.
  Hence, all possible combinations are: ways.
  
  

Hence, all possible cases are



Hence, the required probability is:



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