Dtjytyk

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A coding theory/probability puzzle

Probability of a deck of cards of 8 blue and 5 whiteWhat is the optimal strategy?Probability: Distinguishable vs IndistinguishableWhat is the probability that out of a deck of 16 cards that you will be dealt 2 cards with the same number?Probability question confusionMathematical puzzle - same probability of given card type for any numberUnderstanding Conditional Probability with Intersections of EventsChances of drawing a card of specific colorprobability of an event cards“Fair” random pick with different pool size

3

$begingroup$

I thought of the following problem and I am stuck in solving it.

Suppose there is a deck of 4 cards with 2 red and 2 blue. I pick 2 cards at random and choose 1 and show the other to my friend. With what probability can my friend find the color of my card if we have agreed on a good strategy in advance?

If my friend always guesses the color red, we only fail in this game if I get 2 blue cards. That happens with probability 1/4 and it is the best I can achieve in this case.

However, if the cards are numbered we can do even better, i.e. there is red card 1 and red card 2, blue card 1 and blue card 2. In this case, we can agree that my friend chooses the color of the card I gave him if the number is 1 and flips the color if it is 2. You can check that the only way we can fail is if I get both the red and the blue card of 1. (If I get both cards of the same color I show him the number 1 card. If I get two cards of different colors I show him the number 2 card and we win.) Since the probability of drawing two cards with the number 1 is 1/6, having numbers on the cards clearly helps.

My question is what happens when there are $N$ cards of $N$ colors ($N^2$ in total) and I draw $N$ cards, choose 1 and reveal $N-1$ cards to my friend (in a random order, i.e. the order cannot encode information). What is the strategy that maximizes our probability of winning? How does this differ if cards are numbered or if they are not?

I am interested both in optimal strategies for small $N$, or with asymptotic bounds for large $N$ in both the numbered and unnumbered cases. Could it be that the probability in the numbered case goes to 1 and the unnumbered case is small? This would be very unintuitive!

probability puzzle coding-theory

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edited 18 hours ago

puzzledguy

asked 19 hours ago

puzzledguypuzzledguy

513

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$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I like this question although it seems like a lot of work to solve it. Have you tried some strategies for general $N$?
  $endgroup$
  – Stan Tendijck
  18 hours ago
  

  
  
  
  


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  2
  

  
  
  
  
  
  

  
  $begingroup$
  The probability of drawing both blue cards is $frac{1}{6}$, not $frac{1}{4}$
  $endgroup$
  – Daniel Mathias
  18 hours ago
  

  
  
  
  


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  $begingroup$
  Having the numbers does not help in your two card case. You win $frac 56$ both ways.
  $endgroup$
  – Ross Millikan
  16 hours ago
  

  
  
  
  

add a comment | 

3

$begingroup$

I thought of the following problem and I am stuck in solving it.

Suppose there is a deck of 4 cards with 2 red and 2 blue. I pick 2 cards at random and choose 1 and show the other to my friend. With what probability can my friend find the color of my card if we have agreed on a good strategy in advance?

If my friend always guesses the color red, we only fail in this game if I get 2 blue cards. That happens with probability 1/4 and it is the best I can achieve in this case.

However, if the cards are numbered we can do even better, i.e. there is red card 1 and red card 2, blue card 1 and blue card 2. In this case, we can agree that my friend chooses the color of the card I gave him if the number is 1 and flips the color if it is 2. You can check that the only way we can fail is if I get both the red and the blue card of 1. (If I get both cards of the same color I show him the number 1 card. If I get two cards of different colors I show him the number 2 card and we win.) Since the probability of drawing two cards with the number 1 is 1/6, having numbers on the cards clearly helps.

My question is what happens when there are $N$ cards of $N$ colors ($N^2$ in total) and I draw $N$ cards, choose 1 and reveal $N-1$ cards to my friend (in a random order, i.e. the order cannot encode information). What is the strategy that maximizes our probability of winning? How does this differ if cards are numbered or if they are not?

I am interested both in optimal strategies for small $N$, or with asymptotic bounds for large $N$ in both the numbered and unnumbered cases. Could it be that the probability in the numbered case goes to 1 and the unnumbered case is small? This would be very unintuitive!

probability puzzle coding-theory

share|cite|improve this question

edited 18 hours ago

puzzledguy

asked 19 hours ago

puzzledguypuzzledguy

513

New contributor

puzzledguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  I like this question although it seems like a lot of work to solve it. Have you tried some strategies for general $N$?
  $endgroup$
  – Stan Tendijck
  18 hours ago
  

  
  
  
  


• 
  
  
  

  
  2
  

  
  
  
  
  
  

  
  $begingroup$
  The probability of drawing both blue cards is $frac{1}{6}$, not $frac{1}{4}$
  $endgroup$
  – Daniel Mathias
  18 hours ago
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Having the numbers does not help in your two card case. You win $frac 56$ both ways.
  $endgroup$
  – Ross Millikan
  16 hours ago
  

  
  
  
  

add a comment | 

$begingroup$

I thought of the following problem and I am stuck in solving it.

Suppose there is a deck of 4 cards with 2 red and 2 blue. I pick 2 cards at random and choose 1 and show the other to my friend. With what probability can my friend find the color of my card if we have agreed on a good strategy in advance?

If my friend always guesses the color red, we only fail in this game if I get 2 blue cards. That happens with probability 1/4 and it is the best I can achieve in this case.

However, if the cards are numbered we can do even better, i.e. there is red card 1 and red card 2, blue card 1 and blue card 2. In this case, we can agree that my friend chooses the color of the card I gave him if the number is 1 and flips the color if it is 2. You can check that the only way we can fail is if I get both the red and the blue card of 1. (If I get both cards of the same color I show him the number 1 card. If I get two cards of different colors I show him the number 2 card and we win.) Since the probability of drawing two cards with the number 1 is 1/6, having numbers on the cards clearly helps.

My question is what happens when there are $N$ cards of $N$ colors ($N^2$ in total) and I draw $N$ cards, choose 1 and reveal $N-1$ cards to my friend (in a random order, i.e. the order cannot encode information). What is the strategy that maximizes our probability of winning? How does this differ if cards are numbered or if they are not?

I am interested both in optimal strategies for small $N$, or with asymptotic bounds for large $N$ in both the numbered and unnumbered cases. Could it be that the probability in the numbered case goes to 1 and the unnumbered case is small? This would be very unintuitive!

probability puzzle coding-theory

share|cite|improve this question

edited 18 hours ago

puzzledguy

asked 19 hours ago

puzzledguypuzzledguy

513

New contributor

puzzledguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

$endgroup$

share|cite|improve this question

edited 18 hours ago

puzzledguy

asked 19 hours ago

puzzledguypuzzledguy

513

New contributor

puzzledguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

share|cite|improve this question

edited 18 hours ago

puzzledguy

asked 19 hours ago

puzzledguypuzzledguy

513

New contributor

puzzledguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 19 hours ago

puzzledguypuzzledguy

513

New contributor

puzzledguy is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.

asked 19 hours ago

puzzledguypuzzledguy

513