# Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real world for a coin to also fall on its side which makes a third event ( $P(\text{side}) = 1 - P(\text{heads}) - P(\text{tails})$ ?). How would a mathematician go about calculating the probability of that? Could it be done by simply using respective surface areas or would a proper model be more complex?

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sounds more like a physics problem to me, since you are talking not about the side that is first to touch the ground (maybe you are), but the side that the coin will remain on? This makes it less of a prob-based-on-area problem. But if you were to ask about probability of a certain area touching the ground first, you would need to define the width of a coin, e.g. by 'fair' coin would you mean that it's so wide that it can touch with either of 3 sides with equal prob? Then there is a question of what about the edge between the third and either of the other sides? In limit, you talk of a sphere. – InterestedGuest Oct 3 '11 at 0:52
And in general the part of a coin to touch the ground first is almost always the real edge between one of the faces and the cylindrical surface we usually call the "edge". So that won't help. – Henning Makholm Oct 3 '11 at 0:59
happened to me once. just one day after I saw it happen in a movie. what is the probability of that happening? :D – pootzko Oct 3 '11 at 9:34
This happened to me the other day. well not me, but I was present and witnessed my friend throw a canadian penny at a table, it fell and bounced twice... then landed on its side. After long serious debate, we were wondering what the odds of this happening would be, since none of us had ever witnessed this phenomenon before. – user72127 Apr 11 '13 at 16:26

I covered this question at my Fair Dice column. If you flip your coin onto sticky pitch, it has a much better chance of landing on edge than if you flip it onto a smooth granite surface. The potential to bounce makes a huge difference.

Roughly, you're looking at a 3-state object. Each time it lands, it will lose some energy, and the chance that it will change to a different state varies with the energy. The amount of energy necessary to topple from one state to another will let you build a transition matrix, and then multiply a series of these together to get expected behavior on a given surface.

More is at the site dicephysics. Cylinders of various size were mechanically rolled thousands of times on a variety of surfaces. Results closely followed the Energy State model of my thesis on dice.

Toppling must be considered. Many objects have unstable sides, and will topple away from them. There is a Unistable polyhedron which will always eventually roll to the same face each time it it tossed.

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"unistable polyhedron" - That reminds me... – J. M. Oct 3 '11 at 1:35
@Ed: doesn't it also matter how the coin is thrown? If I "throw" the coin with a "bowling" motion, I would expect a much higher chance of it rolling on an edge and finishing its motion on an edge, for example. – Fixee Oct 3 '11 at 2:55
Yes. If you roll a coin onto sticky pitch, the odds it will land on edge are %100 if you do it right. If you launch it from a suitably powerful launcher, you can embed the coin into wooden surfaces. – Ed Pegg Oct 3 '11 at 3:00

Experiments and models of physical coin flipping are given in:

Diaconis, Holmes, and Montgomery, Dynamical Bias in the Coin Toss comptop.stanford.edu/preprints/heads.pdf

and

Murray and Teare, The probability of a tossed coin falling on its edge, Phys. Rev. E. 2547-2552 (1993)

abstract of 1993 paper: "An experiment is reported in which an object which can rest in multiple stable configurations is dropped with randomized initial conditions from a height onto a flat surface. The effect of varying the object’s shape on the probability of landing in the less stable configuration is measured. A dynamical model of the experiment is introduced and solved by numerical simulations. Results of the experiments and simulations are in good agreement, confirming that the model incorporates the essential features of the dynamics of the tossing experiment. Extrapolations based on the model suggest that the probability of an American nickel landing on edge is approximately 1 in 6000 tosses."

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Oh, Persi Diaconis... what haven't you experimented on? :D – J. M. Oct 3 '11 at 3:03

While I am and always have been a big proponent of the edge factoring into probability in just a quirky enough way as to disrupt an absolute $50-50$ "fair" game and create the equivalent to the least profitable house advantage in existence kind of like $0$ and $00$ on a roulette wheel, the idea that the probability is as good as $1$ in $6000$ or $.0166%$ seems kind of high even for a nickel.

I guess I think about it in terms of if possibility exists in any way, regardless of probability, in the moment of the flip is the coin in a state of fluctuation that is suggestive of one of $2$ outcomes but indeed it is not a definite zero-sum game in that a $3$rd party, as moronic as his speculation may be, could be the benefactor of the outcome should he place his position on a "house" win... I dunno, it's that kind of stuff that distracts me from learning anything productive! :)

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Hi Gonzo. Thank you for sharing your thoughts. For what you have mentionned it could help to think in terms of modelling (see coursera.org/course/modelthinking). For any observed situation there might be hundreds of subtle variables which could prove difficult to determine. What is usually done is to produce a basic model and then gradually improve it until it provides results that are usable for an intended purpose like predicting a phenomenon. – James Poulson Jul 14 '12 at 22:42

of course it depends upon the thickness of the coin , if the thickness is much then the chances of finishing its motion on the side could be infinite coz it has infinite points on its side.

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