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'Snake-Eyes ' Joe (Posted on 2008-07-28) Difficulty: 3 of 5
"Snake-Eyes" Joe introduced a die of his own into a game of chance.

He was subsequently challenged that the die was biased.

Very rigorously test to see if there are grounds to substantiate this claim; don't accept just two or three trial runs. Are you able to offer a theoretical model consistent with your findings?

Test "Snake-Eyes" Joe's Die with this simulator which has a run of 60,000 at a time:

No:123456Total
Scores 0 0 0 0 0 0 0

Note: the data changes with each subsequent mouse-over visitation to the link.

See The Solution Submitted by brianjn    
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Solution a solution | Comment 1 of 13

While the textbook answer to this question would involve the chi-square test applied to as many rolls of the die as possible, the division into groups of 60,000 makes for an easier, but still rigorous, test, whose certainty "high enough", though not as high as the same figures analyzed through chi-square.

Here are 10 runs of 60,000:

9309 10286  9959 10243 10064 10139 60000 
9190 10242 10029 10221 10202 10116 60000
9281 10135 10206 10087 10186 10105 60000
9232 10179 10124 10207 10106 10152 60000
9340 10167 10103 10217 10181  9992 60000
9252 10234 10226 10119 10109 10060 60000
9233 10040 10232 10200 10168 10127 60000
9250 10155  9991 10084 10248 10272 60000
9350 10196 10083 10134 10058 10179 60000
9241 10259 10047 10085 10314 10054 60000

In each of the ten sets of trials the 1 has come up the least often of all the sides of the die.

That one particular face should come up least often in each of the ten sets of trials is likely to happen by chance with probability of only 1 in 10^9.

As mentioned. If we added up each column and applied a chi-square test to the totals, I'm sure we'd have an even less likelihood of this being possibly the result of a uniform distribution. I'll leave that to someone more up on chi-square tests.


  Posted by Charlie on 2008-07-28 16:38:13
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