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In a bar (Posted on 2016-02-23) Difficulty: 3 of 5
N people roll a die in turn, following a prearranged list. The first to get a 6 names the drink. The second to get a 6 drinks it. The third, pays! Then the next on the list rolls and so on till the closing time.

The above goes on for 120 minutes, averaging 5 rolls per minute (the time for naming, drinking, plus settling disputes and paying the bills was discounted).

How many times was it the same player that named the drink, consumed it but did not pay for it?

a. Provide your estimates for N=3, N=6 and N=12
b. Please explain the meaning of the results.

Rem: Verification of analytical results by simulation is welcome.

No Solution Yet Submitted by Ady TZIDON    
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Comments: ( Back to comment list | You must be logged in to post comments.)
re: Last Call | Comment 5 of 6 |
(In reply to Last Call by Steve Herman)

Simulation when the tosses are extended beyond 600 to accommodate an incomplete set:

3 6.7041
4 5.101
5 3.8901
6 3.072
7 2.3909
8 1.9112
9 1.5317
10 1.2545
11 1.0285
12 .8293

 
 ntrials = 10000
 
 For n = 3 To 12
 Randomize Timer
   successct = 0
   For tr = 1 To ntrials
     ct6 = 0: tosser = 0
     toss = 1
     Do
       DoEvents
       tosser = tosser + 1: If tosser > n Then tosser = 1
       rslt = 1 + Int(6 * Rnd(1))
       If rslt = 6 Then
         ct6 = ct6 + 1
         who(ct6) = tosser
         If ct6 = 3 Then
           If who(1) = who(2) And who(2) <> who(3) Then successct = successct + 1
           ct6 = 0
         End If
       End If
       toss = toss + 1
     Loop Until toss > 600 And ct6 = 0
   Next tr
   Text1.Text = Text1.Text & n & Str(successct / ntrials) & crlf
 Next n
 


Theoretical count using 203/6 instead of 100/3:

 3   7975//1183   6.7413355874894336432 
 4   2309125//450241   5.1286422160576224732 
 5   85133125//21631801   3.9355541870970429137 
 6   421496875//137560423   3.0640853365215371574 
 7   3394015625//1404402749   2.4166967968531084098 
 8   3200596328125//1661497798081   1.9263319709611598957 
 9   14601851171875//9429807704863   1.5484781481116259418 
 10   3221265455078125//2570545871703601   1.2531445132092767339 
 11   100509881376953125//98576489633282761   1.0196131121209816896 
 12   444584912060546875//533586283299572503   0.8332015382991818863
 
 
 
    5   open "inabarx.txt" for output as #2
   10   for N=3 to 12
   20    P=((5//6)^(N-1)//6)//(1-(5//6)^N)
   25    Q=1-P
   30    print #2,N,203*P*Q//6,203*P*Q/6
   40   next
   50   close #2


  Posted by Charlie on 2016-02-24 10:22:27
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