- #1
chromomancer
- 2
- 0
I've have just registered here in order to post this question. I know it's a good idea to read a
forum for weeks and months (years!) before daring to post, in case a question has already been
discussed in detail -- but it's a burning question! (And it's not homework.)
If it is off-topic, or has been discussed elsewhere, I'd appreciate a link, but I am looking for a serious discussion of what factors might be involved and how the problem might be tackled (I've studied physics at degree level, but that was many years ago).
Question follows:
What is the probability that a coin will land on its edge?
No, it's a serious question. I know that, assuming Newton's
laws, the behaviour is predictable and it's supposed to
come up heads more often than tails if flipped by a human
being, for example. I'm not interested (here) in that kind
of smart answer. Take the question as intended: you know what
I mean, but I'll spell out some reasonable assumptions
below.
It is usually assumed that there is a small, but non-zero,
chance that it can end standing up on its edge.
If the coin is a perfectly symmetrical flat cylinder, and
dropped vertically (in a vertical orientation) onto an
ideal smooth, flat, hard surface, then since there is
nothing to break the symmetry it must end up on its edge.
But what about a real coin?
Assume it is ejected into the system by a sufficiently
chaotic "coin tossing" mechanism that it starts out
from effectively a random velocity, angular momentum,
position and orientation, and drops onto a smoothish,
fairly hard (but not perfectly smooth) surface and the
bounces are not perfectly elastic. Moreover, the initial
horizontal component of velocity is small, but not zero.
The exact form of the initial random distributions probably
don't matter -- but I'll leave you to decide if they do.
I can see two possibilities: either the initial sideways
motion can never be canceled exactly so the chance of the
coin ending up on its edge are exactly zero (landing on
a carpet, or a sticky surface would be diferent). Or there
really is a small chance that it will stop upright.
My hunch is that it is possible, but the chances are much
less than one in a million. I'm not good (no humans are)
at estimating the probability of very low frequency events
by "common-sense" or intuition. But it may be possible to
do a BOTE calculation to get some idea.
I'd guess it's less than the difference in probability between
heads and tails (real coins are not perfectly symmetrical).
So what I want is an order of magnitude estimate: is it
one in a million, one in a trillion, one in 10^30 or what?
Thanks,
Jonathan
forum for weeks and months (years!) before daring to post, in case a question has already been
discussed in detail -- but it's a burning question! (And it's not homework.)
If it is off-topic, or has been discussed elsewhere, I'd appreciate a link, but I am looking for a serious discussion of what factors might be involved and how the problem might be tackled (I've studied physics at degree level, but that was many years ago).
Question follows:
What is the probability that a coin will land on its edge?
No, it's a serious question. I know that, assuming Newton's
laws, the behaviour is predictable and it's supposed to
come up heads more often than tails if flipped by a human
being, for example. I'm not interested (here) in that kind
of smart answer. Take the question as intended: you know what
I mean, but I'll spell out some reasonable assumptions
below.
It is usually assumed that there is a small, but non-zero,
chance that it can end standing up on its edge.
If the coin is a perfectly symmetrical flat cylinder, and
dropped vertically (in a vertical orientation) onto an
ideal smooth, flat, hard surface, then since there is
nothing to break the symmetry it must end up on its edge.
But what about a real coin?
Assume it is ejected into the system by a sufficiently
chaotic "coin tossing" mechanism that it starts out
from effectively a random velocity, angular momentum,
position and orientation, and drops onto a smoothish,
fairly hard (but not perfectly smooth) surface and the
bounces are not perfectly elastic. Moreover, the initial
horizontal component of velocity is small, but not zero.
The exact form of the initial random distributions probably
don't matter -- but I'll leave you to decide if they do.
I can see two possibilities: either the initial sideways
motion can never be canceled exactly so the chance of the
coin ending up on its edge are exactly zero (landing on
a carpet, or a sticky surface would be diferent). Or there
really is a small chance that it will stop upright.
My hunch is that it is possible, but the chances are much
less than one in a million. I'm not good (no humans are)
at estimating the probability of very low frequency events
by "common-sense" or intuition. But it may be possible to
do a BOTE calculation to get some idea.
I'd guess it's less than the difference in probability between
heads and tails (real coins are not perfectly symmetrical).
So what I want is an order of magnitude estimate: is it
one in a million, one in a trillion, one in 10^30 or what?
Thanks,
Jonathan