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Some time ago, Persi Diaconis was discussing probability and made the point that the level of information a person possessed about an event played a role in his accessing probabilities. To illustrate this point, he did a very nice demonstration showing how a magician can toss a fair coin in such a way that he will always get heads.

(See 2:10 in a video of Diaconis

I am curious if this is possible using a fair die?

TIA, Matt

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Tossing a coin so as to land the predicted way isn't all that hard, it just takes practice. (I've heard of kids tossing coins to determine the start of a soccer match - instead of calling heads/tails, they'd call 'odds/evens' (meaning different side, same side) because everyone could toss the coin predictably. A dice roll is much less predictable than a coin toss, so I'd say no, it's not possible. But given that magicians regularly do things that I'd say were impossible, maybe. It depends on what mechanisms they are allowed to use. – Jeremy Miles Jan 6 '14 at 16:50
You might be interested in Derren Brown's TV programme entitled "The Experiments: The Secret of Luck" if you want to see a version of this trick performed in front of an audience. – Flounderer Jan 6 '14 at 20:32
I was curious recently and tried to find a method to make an "unfair" coin. It turns out to be rather hard to do, where "hard" is defined as "Google can't show evidence that anyone has done it". Worse, Google easily turned up a classroom exercise where students attempt to make loaded or unfair coins then test them with statistics to see if they succeeded. The exercise did fail to even attempt to control for operator skill. – RBerteig Jan 6 '14 at 23:03
up vote 11 down vote accepted

It's absolutely possible. Beyond the trivial example of rolling a die so that it falls out of hand rolling over exactly once, skilled rollers can throw along one rotational axis so that 2 of the 6 faces are eliminated (or simply less likely).

An informative example is casino craps: the person rolling must bounce the dice off the back wall for the throw to be considered legal. This rule was instituted because of actual problems of people biasing their throws; the technique is known as "dice sliding".

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... and I was typing essentially the same answer (+1) in as you hit the post button!! – jbowman Jan 6 '14 at 18:46
This is exactly what I was wondering. I supposed it was possible but didn't know for certain or how it could be done. Thank you for the explanation and informative example! – Matt Brenneman Jan 6 '14 at 20:22

This is more of a physics question, isn't it? Assuming you can control a specific way of rolling well enough, you could get any number you want by simply adjusting the starting position of the die in your hand.

The trivial case would be making a die roll over only once. Almost anyone can do this and thus control the result based on the position of the die before the roll.

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When Diaconis did the coin flip for example, it looked like a genuine coin flip (i.e. it didn't just flip once). I imagine a die would be more difficult to control, but I am am just curious if it can be done – Matt Brenneman Jan 6 '14 at 16:52

The video linked discusses probability of the outcome of a coin-flip as a consequence of the observers knowledge of the world. The magic trick is an illustration.

On the subject of surprising results in the realizations of coin flips.

The $P(TTH)$ occurring first in a run of coin-flips is twice the $P(THT)$ than after it. The $P(THH)$ occurring first in a run of coin-flips is three times the $P(HTT)$ than after it.

There's a more at Mathworld on Coin flips.

Diaconis' more famous work about card shuffling uses group theory and Martingales to study how the entropy of separation distances increases with shuffling. How many times must a deck of cards be shuffled until it is close to random?

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I think your question revolves around how you define stuff.

When I think of a fair die or fair coin I think of an abstract entity that produces a sequence of random results that satisfies its advertised probability distribution.

So a fair die (or count or tack) by definition cannot be biased.

Perhaps another way to look at it is that your question is stated in a way that is bound to lead to a paradox.

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I think you're right to question the definitions, but there is no paradox if we take a "fair" die to be one that when rolled in a standard way has the statistical properties we expect. The point is that one needs to distinguish between a physical object and an experimental setup that yields repeated observations, both of which might be referred to briefly as "a fair die"--whence the need for clear definitions. As Marc Claesen points out, it is not terribly difficult to create an experimental setup with a physically "fair" die that produces a non-uniform distribution of results. – whuber Jan 6 '14 at 17:03
@whuber. It is hard to do and make it appear realistic (you can see Diaconis's video above) – Matt Brenneman Jan 6 '14 at 17:10
@user1172468 The term "fair" depends in part on how the process is performed, which is what Diaconis's point was. As Diaconis shows, an ordinary coin can be flipped in such a way that the outcome is biased (even though the flip "appears" to be fair) because the process of randomization is not (in Diaconis's words) "vigorous" enough. – Matt Brenneman Jan 6 '14 at 17:11
@Matt Rather than data analysis or statistics, you seem to be focusing on physics and magic. We're not the place to be discussing either of those. If there is a form of your question that would be on topic here (consult our help center), then please edit it accordingly. – whuber Jan 6 '14 at 17:24

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