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.
(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
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Posted by Charlie
on 2016-02-24 10:22:27 |
re: Last Call
