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I kinda heard that usual coins are not exactly fair.

The experiment done way ago and one done relative recently could reject the null

hypothesis of p=1/2 after maybe hundreds of thousands of coin tossing!

And I heard the probability of dice is not fair either.

The obvious reason is that the dot mark which is carved into the dice causes

some weight unbalance.

I am curious if anyone know the reference for the experiments I mentioned above?

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There was a pretty funny paper in The American Statistician a few years ago: You can load a die, but you cannot bias a coin. As far as I can recall, they flipped beer bottle caps or some other obvious non-coins, still producing results close to 50-50.

Given the publication bias towards significant results, of any 1000 studies that tossed a coin 1,000,000 times, the 50 that found a significant difference from 0.50 will be published. Meta-analysis can uncover though that 25 would find a positive bias, and 25, a negative bias.

Read about John Kerrich for the real reasons one would want to toss a coin for a few months.

As a class activity, I had my undergrad students sand-paper a few cubes, roll them and prove to me, using Pearson $\chi^2$ test, that they indeed produced a biased die. For the time limits they had (50 to 100 rolls), you had to basically reduce one of the sides to a half to see significant results.

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  • $\begingroup$ This answer is the closest to what I imagined. But not quite so. Here's the reason. I thought the reason that we can reject that p=1/2 is in the same sense that we can almost always reject the null hypothesis given sufficiently large sample size. Even if we built the fair coin possible, and even if we flip it the best way mentioned in the article above, I would bet that we can reject the hypothesis that p=1/2 given very large number of tossing. And I think it doesn't matter if we set alpha=0.05 or 0.01. The reason is that p=1/2, 1/2 exact. It means when p is almost the same as 1/2, but not $\endgroup$
    – KH Kim
    Jun 10, 2012 at 3:05
  • $\begingroup$ quite so, like 0.5000001, or 0.50000000000000001. If p is not exact 1/2, we can spot it with very large sampling. p=1/2 is sort of a unicorn in the same sense that the ideal circle doesn't exist in reality. But I am not sure about this. So I would like to see the experimental result that tossed the allegedly fair coin like zillion times! I heard the first attempt was in some time around the time when probability theory was first invented and recently another attempt was made through zillion times of tossing! Anyway the reference above was also interesting! $\endgroup$
    – KH Kim
    Jun 10, 2012 at 3:11
  • $\begingroup$ Gelman's argument that in a coin toss, you cut the continuous sampling space (angle of the coin to the horizon) in half, so whatever object you are tossing, you will end up with 50-50. I personally think that the argument of whether the probability of heads is exactly 0.5 or 0.5+epsilon is at best a scholastic one. $\endgroup$
    – StasK
    Jun 10, 2012 at 21:19
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Here is a link to a review Diaconis, P., Holmes, S. and R. Montgomery, 2007, “Dynamical Bias in the Coin Toss”, SIAM Review 49, 211–235.

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Just as a personal note. I was a graduate student at Stanford when Persi Diaconis was an assitant professor. Michael Cohen and I were asked to run a dice tossing experiment to see if the dice were fair. We rolled 6 dice on a felt cloth thousands of times. However after the experiment we learned that the dice were shaved and bias favoring two sides over the other four was to be expected. I am not surprised that persi has done many such experiments of this type and published in the reference cited by simmmons. I also remember the coin tossing experiment of Joe Keller (discussed in the cited article) being discussed by Keller and Diaconis back in the late 1970s.

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