How does a coin toss? A look under an asymptotic microscope
HHow does a coin toss ?
Jithin D. George
Abstract — Is flipping a coin a deterministic process or arandom one? We do not allow bounces. If we know the initialvelocity and the spin given to the coin, mechanics should predictthe face it lands on. However, the coin toss has been everyone’sintroduction to probability and has been assumed to be thehallmark random process. So, what’s going on here?
I. P
REFACE
This article is an exploration of the problem describedby Keller in [1]. Keller’s idea serves as inspiration butrestating his proof does not reveal anything different from[1]. So, here, a tangential perspective is explored startingfrom section 4. All the figures are an implementation ofthe theory described here and can be explored through theIPython Notebook [2].II. I
NTRODUCTION
If we know the initial velocity of the coin and its initialangular momentum, it should be a deterministic problem tofind the face it lands on. Let y(t) be the height of the coin attime t , u and w be the initial velocity and the angular velocityimparted to the coin and g be the acceleration due to gravity.We can find the height of the coin and the orientation of thecoin at any time by these equations. y ( t ) = ut − gt + y (0) θ ( t ) = ωt For the rest of this article, we will assume that we start withheads up and returns to the height it started from withoutany bounces. If the time of landing is t , we end up withheads up if nπ − π < θ ( t ) < nπ + π That is, nπ − π < wt < nπ + π We have, from ut − gt = 0 that t = 2 ug nπ − π < w ug < nπ + π Thus, to get heads, w must have the following relation with u . (cid:18) n − (cid:19) πg u < w < (cid:18) n + 12 (cid:19) πg u So, the boundaries separating heads and tails are curves thatsatisfy w = (cid:18) n − (cid:19) πg u , n = 1 , , , . . . u Fig. 1: Curves that separate heads from tails.III. K
ELLER ’ S I DEA
How then is the probability of getting a heads ?Let’s take a look at the curves in the Fig 1. As u getslarge, the curves look like parallel lines that get closer andcloser together. In the large u limit, the lines are so closethat it becomes impossible to pick out heads and tails.
20 22 24 26 28 30 32 u Fig. 2: The curves for large u look like straight lines. a r X i v : . [ phy s i c s . c l a ss - ph ] A p r eller gives a rigorous and beautiful proof for why theprobability becomes as u → ∞ in the appendix of [1]. Inthe next section, we explore it with a different ideology.IV. A LACK OF ’ CONTROL ’If we were able to control our u and w with infiniteprecision, no matter how close the curves get, we wouldbe able predict exactly if we would get a heads or a tails.Let us aim for an initial velocity of u and make an errorof magnitude ∆ u in imparting it. Then, the boundary curvesbecome w = (cid:18) n − (cid:19) πg u = (cid:18) n − (cid:19) πg u + ∆ u ) ∼ (cid:18) n − (cid:19) πg u (1 − ∆ uu ) Our approach is to look at the variation of ω with thechange in u . Since u = u + ∆ u where u is fixed and ∆ u varies, the ω vs. u can be interpreted as the ω vs. ∆ u curveand vice versa.For very large u , we have w ∼ (cid:18) n − (cid:19) πg u (cid:18) − ∆ uu (cid:19) (1)Then, ω has a straight line relation with the error that wemake ( ∆ u ). This makes sense because we know that for large u , the w vs. u curve is a straight line. w = c + m ∆ u One would think the slope of such a line is (cid:18) n − (cid:19) πg u This is not quite right. If n is small compared to u , thiswould mean a slope of 0 corresponding to the region aroundthe blue dot in Fig 3.. If n is very large compared to u ,this would mean a slope of infinity corresponding to the reddot. This is why asymptotics have to be done with a carefulhandle on reality. What we want is the slope near the greendot which corresponds to the straight line approximations.By symmetry, we find that the slope is − . u Fig. 3: Asymptotic approximations need to make senseFinding the slope serves as a nice exercise to remindourselves where we are working on but what we really needfor the rest of this paper is the offset distance between thesuccessive lines. That distance a , shown in Fig 4 and Fig 5.is given by a = πg u This allows us to ask the question: What is the maximumpermissible error one can make in imparting u and w to thecoin? The answer to this is the largest square centered around u such that it lies in the region between the two lines.
19 20 21 22 23 24 25 26 u a Fig. 4: Lying inside this black region allows you to get headswith absolute certainty. Note that only 4 curves have beenshown here for better clarity.The diagonal of the square is given by d = a √ πg √ u and the sides are ∆ u = ∆ w = d √ πg u The ∆ u and ∆ w defined above tell the minimum level ofcontrol we need to have to make the coin toss deterministic.An inability to control it at this level leads to randomness.. D IAGONAL DEFINITIONS
The diagonal of the square in the previous section wasnamed d . This is done because we start with some u ,choose the two closest parallel lines on either side of u andthen find the largest square that fits in between the lines.
19 20 21 22 23 24 25 26 u d a Fig. 5: d is the diagonal of the largest square that lies withinthe two adjacent lines on either side of a particular u .We can choose the second closest parallel lines on eitherside of u , find the largest square that lies in between themand refer to its diagonal as d . Similarly, we can find d n diagonal for larger and larger squares defined this way. d , d and d are highlighted in Fig 6.
20 21 22 23 24 25 26 u d d d Fig. 6: d and d can be found in a similar way to d .It is not too difficult to show that the n th diagonal lengthis given by d n = (2 n − πg √ u VI. A
BIGGER SQUARE
If we increased the size of square further, the probabilityof getting heads would decrease from 1.
19 20 21 22 23 24 25 26 u ABB
Fig. 7: When the square of ‘control’ gets outside region A,the coin toss becomes random.Let’s first consider the case when the square is outside re-gion A but still inside region B. So, the probability decreasesmonotonically with the size of the square. Using the areasof the two regions, we can calculate the probabilities.Let d be the diagonal of the square. Clearly, d < d < d .The total area of the square is d Also, the total area of each of the two triangles in region Bis
14 ( d − d ) . So, the combined area of the two triangles is
12 ( d − d ) Hence, the probability of tails is given by the ratio of theareas. P T = d − d d and that of heads is given by P H = 1 − d − d d u * P H Fig. 8: Probability of heads as the square gets into region B.n general, we wish to understand the probability distri-bution for situations like in the Fig 7 below.
24 25 26 27 28 29u242526272829 a b b b b c B c
Fig. 9: A more complicated square.Area of region a = 12 d −
12 ( d − d ) Area of region b n = 2 d − d n − − d n d Area of region c = 14 ( d − d N ) We can use these areas to find the probability of getting aheads as the size of the square increases. It is shown in Fig7. u * P H Fig. 10: Probability of heads as the error (normalized) in uincreases.Fig 8 allows us to see how as you lose control overthe impulse imparted to the coin, the probability of headseventually goes to 0.5. Interestingly, the probability nevergets below 0.5. This means that you are ever slightly morelikely to get the side you started with when you tossed thecoin. Persi Diaconis found a similar bias in the toss of ausual coin in [3]. The probability of heads in that study was0.51! (Not a factorial, an exclamation). VII. R
EAL LIFE SCENARIOS
From personal experimentation, a coin is tossed with avelocity near 4m/s which does not qualify it to be in the u →∞ region. Furthermore, measuring ω is extremely difficultas expressed by Diaconis in his Numberphile videos.VIII. C ONCLUSIONS
This article tries to explore how the meaning of whatwe call random or deterministic is related to how muchcontrol we have over it. It also hopefully leaves readers withthe impression that they can never look at the simple cointoss the same way again. Something we always thought wasrandom could be deterministic based on our level of control.This insight also serves as a cure for one of the dilemmasfaced by people introduced to statistical mechanics. Themotion of particles is described by Newton’s laws and henceshould be completely deterministic. That motion howevertakes place at such a small scale and with such a largenumber of interactions that our lack of control or ‘precision’makes it a random process.R
EFERENCES[1] Keller, Joseph B. ”The probability of heads.” The American Mathemat-ical Monthly 93.3 (1986): 191-197.[2] IPython notebook : https://github.com/Dirivian/Jupyter_notebooks/blob/master/Coin_toss.ipynbhttps://github.com/Dirivian/Jupyter_notebooks/blob/master/Coin_toss.ipynb