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Perfect Pairs (Posted on 2009-02-11) Difficulty: 4 of 5
Suppose the local casino introduces a card game named "Perfect Pairs". The game involves a player making a bet followed by a dealer dealing out two cards to every player who has made a bet. If the player has a pair, the player wins 11 times his initial bet as well as keeping his initial bet. If the player does not get a pair, the player loses all the money he has bet in that round to the casino.

Additionally, suppose that six full decks of cards are initially shuffled and used and the dealer does not re-shuffle the cards until 5 decks of cards are used up. For the sake of terminology we will call a set of rounds that are played without the cards being shuffled a "match".

If there are n players always playing the game, what is the expected percentage of "matches" that will have at least one round in which a player who has memorized the previous cards dealt in that "match" could calculate that he has an edge over the dealer (ie, expected percentage of matches in which there is at least one round in the match when the chance of making a pair exceeds 1/12)?

No Solution Yet Submitted by Chris, PhD    
Rating: 3.0000 (2 votes)

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Some Thoughts simulation | Comment 1 of 6

This VB 5 program simulates matches with given numbers of players.

Dim denCt(13), totCards, hit, totHits, tries


Private Sub cmdStart_Click()
 For n = 1 To 5
    tries = 0: totHits = 0
    Do
     DoEvents
     For i = 1 To 13
      denCt(i) = 24
     Next
     totCards = 52 * 6
     hit = 0
     Do
       For i = 1 To 2 * n
        dealCard
       Next
       If totCards > 52 Then
        prob = 0
        For i = 1 To 13
         prob = prob + denCt(i) * (denCt(i) - 1) / (totCards * (totCards - 1))
        Next
        If prob > 1 / 11 Then hit = 1
       End If
     Loop Until totCards <= 52
     totHits = totHits + hit
     tries = tries + 1
    Loop While totHits < 5000
    Print Str(totHits) & Str(tries) & Format(totHits / tries, " 0.00000")
 Next
End Sub


Private Sub Form_Load()
 Randomize
End Sub

Sub dealCard()
  DoEvents
  rNo = Int(Rnd(1) * totCards + 1)
  If rNo > totCards Then rNo = totCards
  rns = rNo
  i = 1
  Do While rNo > denCt(i)
   rNo = rNo - denCt(i)
   i = i + 1
  Loop
  denCt(i) = denCt(i) - 1
  totCards = totCards - 1
End Sub


Runs with 1 through 5 players found:

players  needed matches for 5000 hits   for this probability per match
   1             381474                        0.01311
   2             547593                        0.00913
   3             462256                        0.01082
   4             632347                        0.00791
   5            1142136                        0.00438
      

One would think that the probability would be monotonically decreasing, so the rise for 3 players would be the result of chance.      


  Posted by Charlie on 2009-02-11 16:41:28
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