Comment author:ksvanhorn
22 January 2011 06:06:27AM
16 points
[-]

"Is it not true that when one rolls a fair 1d6, there is an actual 1/6 probability of getting any one specific value?"

No. The unpredictability of a die roll or coin flip is not due to any inherent physical property of the objects; it is simply due to lack of information. Even with quantum uncertainty, you could predict the result of a coin flip or die roll with high accuracy if you had precise enough measurements of the initial conditions.

Let's look at the simpler case of the coin flip. As Jaynes explains it, consider the phase space for the coin's motion at the moment it leaves your fingers. Some points in that phase space will result in the coin landing heads up; color these points black. Other points in the phase space will result in the coin landing tails up; color these points white. If you examined the phase space under a microscope (metaphorically speaking) you would see an intricate pattern of black and white, with even a small movement in the phase space crossing many boundaries between a black region and a white region.

If you knew the initial conditions precisely enough, you would know whether the coin was in a white or black region of phase space, and you would then have a probability of either 1 or 0 for it coming up heads.

It's more typical that we don't have such precise measurements, and so we can only pin down the coin's location in phase space to a region that contains many, many black subregions and many, many white subregions... effectively it's just gray, and the shade of gray is your probability for heads given your measurement of the initial conditions.

So you see that the answer to "what is the probability of the coin landing heads up" depends on what information you have available.

Of course, in practice you typically don't even have the lesser level of information assumed above -- you don't know enough about the coin, even in principle, to compute which points in phase space are black and which are white, or what proportion of the points are black versus white in the region corresponding to what you know about the initial conditions. Here's where symmetry arguments then give you P(heads) = 1/2.

Comment author:bigjeff5
01 February 2011 06:14:41PM
6 points
[-]

Case in point:

There are dice designed with very sharp corners in order to improve their randomness.

If randomness were an inherent property of dice, simply refining the shape shouldn't change the randomness, they are still plain balanced dice, after all.

But when you think of a "random" throw of the dice as a combination of the position of the dice in the hand, the angle of the throw, the speed and angle of the dice as they hit the table, the relative friction between the dice and the table, and the sharpness of the corners as they tumble to a stop, you realize that if you have all the relevant information you can predict the roll of the dice with high certainty.

It's only because we don't have the relevant information that we say the probabilities are 1/6.

Comment author:Juno_Watt
21 May 2013 05:27:32PM
0 points
[-]

If you knew the initial conditions precisely enough, you would know whether the coin was in a white or black region of phase space, and you would then have a probability of either 1 or 0 for it coming up heads.

Not necessarily, because of quantum uncertainty and indeterminism -- and yes, they can affect macroscopic systems.

The deeper point is, whilst there is a subjective ignorance-based kind of probability, that does not by itself
mean there is not an objective, in-the-territory kind of 0<p<1 probability. The latter would be down to
how the universe works, and you can't tell how the universe works by making conceptual, philosophical-style arguments.

So the kind of probability that is in the mind is in the mind, and the other kind is a separate issue. (Of course, the existence of objective probability doesn't follow from the existence of subjective probability any more than its
non existence does).

## Comments (186)

Old"Is it not true that when one rolls a fair 1d6, there is an actual 1/6 probability of getting any one specific value?"

No. The unpredictability of a die roll or coin flip is not due to any inherent physical property of the objects; it is simply due to lack of information. Even with quantum uncertainty, you could predict the result of a coin flip or die roll with high accuracy if you had precise enough measurements of the initial conditions.

Let's look at the simpler case of the coin flip. As Jaynes explains it, consider the phase space for the coin's motion at the moment it leaves your fingers. Some points in that phase space will result in the coin landing heads up; color these points black. Other points in the phase space will result in the coin landing tails up; color these points white. If you examined the phase space under a microscope (metaphorically speaking) you would see an intricate pattern of black and white, with even a small movement in the phase space crossing many boundaries between a black region and a white region.

If you knew the initial conditions precisely enough, you would know whether the coin was in a white or black region of phase space, and you would then have a probability of either 1 or 0 for it coming up heads.

It's more typical that we don't have such precise measurements, and so we can only pin down the coin's location in phase space to a region that contains many, many black subregions and many, many white subregions... effectively it's just gray, and the shade of gray is your probability for heads given your measurement of the initial conditions.

So you see that the answer to "what is the probability of the coin landing heads up" depends on what information you have available.

Of course, in practice you typically don't even have the lesser level of information assumed above -- you don't know enough about the coin, even in principle, to compute which points in phase space are black and which are white, or what proportion of the points are black versus white in the region corresponding to what you know about the initial conditions. Here's where symmetry arguments then give you P(heads) = 1/2.

Case in point:

There are dice designed with very sharp corners in order to improve their randomness.

If randomness were an inherent property of dice, simply refining the shape shouldn't change the randomness, they are still plain balanced dice, after all.

But when you think of a "random" throw of the dice as a combination of the position of the dice in the hand, the angle of the throw, the speed and angle of the dice as they hit the table, the relative friction between the dice and the table, and the sharpness of the corners as they tumble to a stop, you realize that if you have all the relevant information you can predict the roll of the dice with high certainty.

It's only because we don't have the relevant information that we say the probabilities are 1/6.

Not necessarily, because of quantum uncertainty and indeterminism -- and yes, they can affect macroscopic systems.

The deeper point is, whilst there is a subjective ignorance-based kind of probability, that does not by itself mean there is not an objective, in-the-territory kind of 0<p<1 probability. The latter would be down to how the universe works, and you can't tell how the universe works by making conceptual, philosophical-style arguments.

So the kind of probability that is in the mind is in the mind, and the other kind is a separate issue. (Of course, the

existenceof objective probability doesn't follow from the existence of subjective probability any more than itsnonexistence does).