To rig coins and cubes
Sometime back, I put up a couple of questions at mathoverflow and they got deleted. I believe the questions deserve to survive.
1. Given a fraction α > 1/2, how does one make a coin for which probability of landing heads in a fair toss is α ?
Basically, we need to 'rig' a coin to a specified extent. By coin, we mean a cylinder with low but non-zero height (compared to radius). Obviously, the density should vary within the body of the coin. The radius to height ratio could be chosen suitably so as to achieve the required α . I am not sure what assumptions should be made of the ground on which the coin lands.
2. How does one 'rig a cube' such that the six faces have some specified set of six probabilities on a toss - for example, 5 of the faces could have equal probability but the sixth has a slightly greater probability. This seems a clearer question in that there is no confusion about the dimensions of the object.
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The questions (espcially the one on coins) had elicited some comments:
- "Fair coins do not really give probability 1/2 for heads. They actually tend to land on the side they started with."
(but as far as I can make out, if one performs the same number of tosses with heads and tails up, this bias could get cancelled).
- "People commonly expect that coin tossing would be mainly affected by the weighting (perhaps just the location of the mass center). Not so; E. T. Jaynes wrote about how the method of tossing is more crucial. He devised three different tossing methods that lead to quite different heads probabilities for the same coin."
(but again, for a fair coin, these biases all seem cancellable by following whatever is the 'complement' of each biased tossing method).
On the whole, my feeling is that for a uniform and equally weighted coin, there are indeed ways of tossing to ensure that as the total number of tosses goes to infinity, the probability of heads does get to 1/2. Then question 1 could be rephrased: how does one alter a coin in such a way that the very manner of tossing that yields 1/2 for heads for a fair coin would yield α probability for heads with this new coin?
1. Given a fraction α > 1/2, how does one make a coin for which probability of landing heads in a fair toss is α ?
Basically, we need to 'rig' a coin to a specified extent. By coin, we mean a cylinder with low but non-zero height (compared to radius). Obviously, the density should vary within the body of the coin. The radius to height ratio could be chosen suitably so as to achieve the required α . I am not sure what assumptions should be made of the ground on which the coin lands.
2. How does one 'rig a cube' such that the six faces have some specified set of six probabilities on a toss - for example, 5 of the faces could have equal probability but the sixth has a slightly greater probability. This seems a clearer question in that there is no confusion about the dimensions of the object.
------
The questions (espcially the one on coins) had elicited some comments:
- "Fair coins do not really give probability 1/2 for heads. They actually tend to land on the side they started with."
(but as far as I can make out, if one performs the same number of tosses with heads and tails up, this bias could get cancelled).
- "People commonly expect that coin tossing would be mainly affected by the weighting (perhaps just the location of the mass center). Not so; E. T. Jaynes wrote about how the method of tossing is more crucial. He devised three different tossing methods that lead to quite different heads probabilities for the same coin."
(but again, for a fair coin, these biases all seem cancellable by following whatever is the 'complement' of each biased tossing method).
On the whole, my feeling is that for a uniform and equally weighted coin, there are indeed ways of tossing to ensure that as the total number of tosses goes to infinity, the probability of heads does get to 1/2. Then question 1 could be rephrased: how does one alter a coin in such a way that the very manner of tossing that yields 1/2 for heads for a fair coin would yield α probability for heads with this new coin?
posted by R.Nandakumar @ 10:39 PM
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