The physics of coin flips

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The physics of coin flips

• Consider a well-balanced circular coin of radius
  a which is tossed upwards, rotates around its
  axis several times, and falls down into wet sand.
• Ignore air resistance.
• Initial height a. Heads side up.
• Initial vertical velocity u, rotational velocity
  w.
• y(t) location of center of gravity
• ?(t) angle between surface normal and heads side
  normal.

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?(t)
y(t)
t
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Equations

• Equtions of motion
• Initial values
• Solution
• When does the coin land?
• When does it land heads?

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More precisely

• At the end points y(t0) a aut0-gt02/2 so
  t02u/g.
• Thus .

10
w
0
0
10
u/g
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What are resonable values?

• u2.4 m/s (determined from maximum height of
  tosses)
• w38 rev/s238.6 rad/s (obtained from wrapping
  dental floss around the coin)
• n19 rev/toss (nwt0)
• Hence u/g0.25 w238.6
• EXTREMELY sensitive to initial conditions.
• Predictable precisely when initial conditions
  well determined.

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But why is P(heads)1/2?

• Option 1 Random side up.
• Option 2 Draw the initial velocities from a
  joint distribution, and make at least one of them
  large. Then one can prove that the probability of
  heads goes to 1/2.

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Weldon rolling dice

• 26 306 rolls of 12 dice
• Frequency of 5 or 6 0.3376986
• Are the dice fair?
• Standard error if fair 0.00084.
• 5 standard errors above expected.
• So why are they not fair?

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Stochastic processes

• Classical model
• General model
• How do you prove that?

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More generally

• S state space
• T time
• Events are measurable subsets of S. These are
  things we can assign probabilities to.

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What is time?

• 1. Xtearthquakes of magnitude gt5 near Mount St.
  Helens in time (0,t
• Snatural numbers N, Tpositive halfline R
• 2. Xk(Bk,Dk)births and deaths in a population
  in year k
• SN2 TN
• 3. Xs,tconcentration of CO at location s, time t
• SR TR2xR
• 4. Xt thickness of fibre distance t from origin.
• STR

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Finite-dimensional distributions

• If time is continuous we cannot write down the
  simultaneous distribution of X(t) for all t.
• Rather, we pick n, t1,...,tn and write down
  probabilities like
• They are called the finite-dimensional
  distributions(fdds) for the process X.

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Kolmogorovs consistency theorem

• Fdds must satisfy the following two conditions
  (sec. 8.6)
• (i)
• as
• (ii)
• for any permutation ? of 1,...,n
• Then a probability space and a stochastic process
  exists with these fdds.

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Purposes of stochastic models

• Aids understanding of the phenomenon
• Hematopoietic stem cells
• More versatile than deterministic models
• Blowfly population model
• Allows assessment of variability
• Long range transport of atmospheric pollutants
• Extension of deterministic models
• Stochastic cloud models

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Group exercise

• For 0t1ltlttn real and 0r1rn integers, define
  X(t) by X(0)0 and
• Find P(X(t)0)
• (b) Determine P(X(t)k)
• (c) Show that X(t1) and X(t2)-X(t1) are
  independent
• (d) Show that Kolmogorovs consistency condition
  is satisfied