Analysis of a Tossed Coin: An Introduction to Phase Space

What does it mean to say that a coin, when tossed, has a 50% probability of landing heads? Since physics is deterministic, depending on the initial configuration of the coin, won't it have either a 100% chance, or a 0% chance? How does the physics behind a tossed coin work anyway? This post will hopefully answer all of these questions.

TL;DR

The information that an observer has about a physical system takes the form of knowing that a system is in a certain region of phase space, containing many points which will evolve differently according to deterministic laws of physics, and the probability of an event is the fraction of phase-space taken up by points which will evolve into that event.

Don't worry if that didn't make sense, the rest of this post is devoted to explaining it.

This post is an extension and adaptation of Chapter 10 of E. T. Jaynes' Probability Theory: The Logic of Science, but it is designed to be accessible to readers who have never heard of the book before.

How to describe a coin

Imagine a coin in the air, above a flat plane which serves as the floor. How many numbers would you need to uniquely describe it? The position would need 3 numbers, one for each of the 3 dimensions of space, x, y, and z. Move the sliders around on the diagram to see how the coin moves when you alter the values of x, y, and z.

FIG. 01Move the coin through three-dimensional space.

What about its orientation? First, we need to know where the heads face of the coin is pointing. Imagine an arrow pointing out of the heads face of the coin (called a normal vector). Describing the orientation of that arrow requires two numbers, one for how close it is to pointing straight up, and one for how it is oriented around the horizontal plane. For those who know, those are the spherical angles $θ$ and $φ$ .

FIG. 02Tilt the coin and change the direction of its face normal.

Is this enough to determine the orientation of the coin? Not quite, imagine fixing the arrow pointing out of the face in the above diagram, and then spinning the coin around that arrow. This is an extra number, $ψ$ , and with it, we have the full orientation of the coin. Given those three numbers, there is only one possible way the coin can be oriented.

FIG. 03Spin the coin around its fixed face normal.

With these 6 numbers, we can uniquely describe the position and orientation of a coin in 3-dimensional space.

FIG. 04Combine position and orientation.

Velocity

Unfortunately, we aren't quite finished. If we want to be able to simulate the coin, to know what it is going to do next, we also need the velocity. Every parameter from before needs a velocity, to determine how fast that parameter is changing. The velocities of the position coordinates ( $x, y, z$ ) tell you how fast the coin is moving, and the velocities of the orientation coordinates ( $θ, φ, ψ$ ) tell you how fast the coin is spinning. This doubles our total parameter count, to 12, $x, y, z, θ, φ, ψ, v_{x}, v_{y}, v_{z}, v_{θ}, v_{φ}, v_{ψ}$ .

FIG. 05Set position, orientation and their rates of change; simulate the resulting motion.

Phase Space

If we take the list of 12 numbers that describe the coin, it is possible to interpret them as coordinates of a 12-dimensional space. You're already familiar with these for smaller dimensions, as the $(x, y, z)$ coordinates of the coin pick out its location in 3-dimensional space. Similarly, you could imagine an abstract "velocity-space", with coordinates $(v_{x}, v_{y}, v_{z})$ . This space wouldn't correspond to where the coin actually was, but how fast it was going. A slow coin would be represented by a point in this space close to the origin, and as you sped up the coin in each direction, it would move out along the corresponding axis in the velocity-space. The name for a generalised space like this, that contains one dimension for every number describing a system, is "phase space", and the phase space of the coin is 12 dimensional, due to the 12 parameters. Working with a 12-dimensional space is impractical, but we can cut it down.

Symmetry

These 12 parameters are annoying to work with, but luckily, if all we care about is simulating a coin to see if it lands heads, we don't need all of them! This is due to the symmetry of the situation. It doesn't matter what the x or y position of the coin is, as moving it in either of those directions cannot change the face that comes down. Similarly, the orientation of the coin around the axis pointing out of the heads face cannot matter, and neither can how that face is oriented in the horizontal plane. None of the velocities of these irrelevant variables can matter either (assuming no wobble of the coin and no air resistance). This knocks out 8 parameters, leaving us with only $z, v_{z}, θ, v_{θ}$ , where $θ$ is the angle from the arrow pointing out of the heads face to the vertical axis, such that $θ = 90^{°}$ is a vertical coin, on its edge, while $θ = 0^{°}$ and $θ = 180^{°}$ are horizontal coins, heads up and tails up respectively. These are all vertical parameters, as the situation is completely horizontally symmetric, moving or rotating the coin horizontally cannot change whether it comes down as heads or tails.

Classifying outcomes

If we think about the condition for a coin landing heads, it has to be facing heads up when the coin hits the floor (for the sake of convenience, I'm taking this to mean when $z = 0$ , which ignores that if the coin is positioned vertically, its edge would touch the floor before it. I am also ignoring bouncing when the coin hits the floor, and air resistance). This means that when $z = 0$ , $0^{°} < θ_{t} < 90^{°}$ . If the coin is tails, then when $z = 0, 90^{°} < θ_{t} < 180^{°}$ . If it takes a certain amount of time $t$ to hit the floor, then the value of $θ$ at that time (denoted $θ_{t}$ ) will be $θ_{t} = θ + t v_{θ}$ , as it starts at $θ$ , and changes by $v_{θ}$ every second. This means that if we know the value of $t$ , $θ$ , and $v_{θ}$ , we can find out whether the coin will come down heads or tails by a simple calculation.

How can we find $t$ ? Expand the derivation if you're interested, but the only important part is that $t$ depends only on $z$ and $v_{z}$ .

Derivation: finding the flight time

The expression giving the height of the coin at landing time is $z + v_{z} t + \frac{1}{2} g t^{2}$ , where $g$ is the acceleration due to gravity. We know that at landing time, $z = 0$ , so $z + v_{z} t + \frac{1}{2} g t^{2} = 0 .$ Solving for $t$ , this yields $t = \frac{- v_{z} - \sqrt{v_{z}^{2} - 2 g z}}{g} .$ More technical readers should re-derive why I selected the negative branch here, given that, realistically, $z > 0, g < 0$ .

We can visualise the dependence of the outcome (heads or tails) on the initial variables like so:

FIG. 06How the initial conditions determine the outcome.

Now, we can graph which values of these parameters lead to which outcomes. In this graph, I've put as axes $t$ and $v_{θ}$ , and coloured regions that lead to heads in yellow, and regions that lead to tails in black. Unfortunately, a 3-dimensional plot isn't so nice on a 2-dimensional screen, so I've put a slider so that you can alter $θ$ .

FIG. 07The outcome map: change the initial angle.

Looking at this graph, and how it changes with the $θ$ slider, it's a valuable exercise to try to convince yourself that this shape makes sense, that is the only shape it could have been. This will require a little mathematical reasoning as to what happens to $θ$ , the angle the heads face of the coin makes with the vertical, over time.

Why the outcome regions form curved bands

The total angle the coin has rotated through at the time of landing $t$ is $t v_{θ}$ , as gravity never changes the rate of rotation. The final orientation of the coin then depends on the remainder when this quantity is divided by 360 degrees, a periodic quantity. This is why there are periodic ribbons of heads-regions and tails-regions when you increase $t$ . These ribbons also have to be symmetric between the $t$ axis and $v_{θ}$ axis, as those variables play symmetric roles in the equation, and moving the $θ$ slider moves them back and forth because altering the initial orientation changes which amounts of total angular rotation correspond to heads and which to tails. The stripes are curved and are shaped like hyperbolas because to maintain a constant orientation while varying $t$ and $v_{θ}$ , you need to move along the shape " $constant = t v_{θ}$ ", which is a hyperbola in the $t, v_{θ}$ plane.

The world and our minds

In the real world, any specific coin corresponds both to a single, exact point in the full 12-dimensional phase space, and a point (plus $θ$ slider position) in the reduced phase space that I've been plotting. Every point in these phase spaces is only heads or tails, a definite outcome.

FIG. 08Each point is a definite outcome. Drag the point to investigate.

However, in our own minds, just looking at a coin, we may not know this exact point. In a standard coin flip, we know that the coin starts horizontally ( $θ$ is around 0 or 180), and have a basic idea of how far it starts above the floor, but we don't know precisely how fast the coin was flicked up, or how fast it is spinning. All we know is that we are in a certain region in the phase space. Exactly where we are in that region is uncertain.

FIG. 09Drag on the plot to change the region of uncertainty.

If we are relatively more certain about a parameter, the width of the region in that parameter is narrow, concentrated only to certain values. If we are less certain, the width of the region would be broad, covering many values (really, what we have is a probability distribution over values, not just a region, as some are more likely than others, but for the sake of simplicity I assume that our distributions are uniform). Now, within that region, there will contain both heads-regions and tails-regions. Because we don't know where we are in the region, we can say that the probability of heads is the area of the contained heads-regions, divided by the area of the total region, and likewise for the probability of tails. As the areas of the two regions add up to the total (the borderlines between them, corresponding to the coin landing on its edge, have 0 area, as we simplify to a perfectly flat coin), we can see that the probability of heads and the probability of tails have to add to 1. If we were including more parameters, we would use higher-dimensional analogues to area, like volume in 3 dimensions.

Some simple examples

Lets zoom in on some regions of phase space, and pretend that we know that the coin starts in specifically that region, but not where in the region.

FIG. 10Equal phase-space area: roughly 50% heads.

If all we knew about the coin was that it was in this region, our probability of it landing heads would be 50%, as it takes up half the region.

FIG. 11An uneven division: approximately two-thirds heads.

In this plot, more like 2/3 of our region of uncertainty is covered by heads, so we say that our probability of it landing heads is 2/3.

FIG. 12A small region entirely inside heads.

If we knew for certain the coin started off in this region, our probability of it landing heads would be 100%, as the region is totally covered by heads.

A realistic example, and what makes coin tossing unpredictable

Let's take some realistic values for uncertainty in $t$ and $v_{θ}$ , assuming that the coin starts off as heads up ( $θ = 0^{°}$ ). The time for a typically flipped coin to hit the ground is between roughly 0.44 and 0.56 seconds, and it rotates vertically between 36 and 40 times per second, going 360 degrees each time, so $v_{θ}$ is roughly between 12,960 and 14,400 degrees per second source. Let's plot this region on our graph.

FIG. 13A typical flip: zoom into the range of plausible initial conditions.

You might be confused as to where the stripes have gone. I recommend zooming in to see them again. The range of values that our uncertainty spans is wide, especially in the $v_{θ}$ direction, and whether the coin lands heads or tails is sensitive to it. In order to land in a region with only a single colour, we would have to flip the coin to get extremely precise values of $t$ and $v_{θ}$ , which is very hard for a human. As we cannot get this kind of control, the coin flip is unpredictable, with a 50% chance of heads.

Why 50%?

The probability of the coin landing heads depends on the region of phase space we know the coin to be in, so to determine that a typical flip has a 50% chance of landing heads, we have to show that equal phase space area from the standard region of uncertainty is devoted to heads and tails. We can establish this from the symmetry of the situation for heads and tails. If you start with a heads-up region, and alter the flight time or spin speed just so slightly to move into a tails-up region, as long as the values of flight time or spin speed are large compared to the change required, the two regions will look almost identical, as all parameters are almost exactly the same. As the human state of knowledge is so broad compared to the widths of these adjacent ribbons of heads and tails in phase space, it must cover a lot of ribbons, which alternate heads and tails with almost exactly the same shape and area, so they take up 50% of the area each. However, there are some regions of phase space where this is not true, and we can get radically different probabilities.

Unfair flips

In regions close to the origin that represent not much time passing or the coin flipping over slowly, the regions of heads or tails are actually quite wide. Imagine seeing someone flip a coin, but they do it close to the floor, so that the coin barely has time to spin, or they don't spin it very fast, essentially just tossing it into the air flat. You would intuitively consider these flips to be unfair. Try to connect this intuition to the graph with wide regions next to the origin.

Why a deliberately slow flip can be unfair

Because the regions of heads and tails are so wide next to the origin, with tiny times or angular velocities, it is easy for a flipper to steer the coin into one of the regions, to get a small region of uncertainty as to how they've spun it. In order for us to view a coin flip as fair, it has to be evident that the distance on the graph that reverses the outcome is much smaller than the smallest distance a flipper can control.

Probability is in our minds

Different people, looking at the coin flip, will have different regions of uncertainty about the properties of the coin. The flipper has a slightly smaller region than someone watching, as they know with more detail how hard they flipped the coin. A robot with a high-speed camera could know the properties of the coin in extreme detail, corresponding to a tiny region, which is likely to reside entirely in a heads-region or tails-region, so that robot would likely be able to predict the coin flip with certainty. As the probability of landing heads depends on the exact region of uncertainty, which varies between people, probabilities can also vary between people. Different observers with different information have different probabilities for the same coin landing heads. This doesn't mean that you can just declare your probability to be whatever you want, there is a correct probability for you to hold derived from the information you have, even if there isn't a single correct probability amongst all observers with all possible different pieces of information.

Conclusion

Returning to the original questions, if we knew exactly how a coin was flipped, then from our perspective it would have either a 0% or 100% chance of landing heads. Our ignorance about the coin landing, represented in our probability of 50%, stems from our ignorance about the coin's initial conditions, and it is essential to the fairness of the flip that the final result of the flip is extremely sensitive to those conditions, to prevent the outcome from being predictable.