# Convergence of an Ergodic process

I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% heads and 33.3...3% tails.

http://ideone.com/v4YTen

Question In the Coinland all the coins in the land were flipped and allowed to fall to the ground. (uniformly heads or tails, no bias coins or funny stuff etc).

As the people in Coinland walked around, if they encountered a coin on the ground they will do one of the following:

1. Flip the coin it if is a tail
2. Turn the coin over if it is a head

After a sufficiently large amount of time has passed, what percentage of the coins on the ground are heads?

I was hoping someone could provide some guidance/suggestions to possible avenues of investigations - if possible please don't provide an exact answer.

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Hint: consider two independent sequences $(\epsilon_n)$ and $(\epsilon'_n)$ of independent random variables uniformly distributed on $\{-1,+1\}$. The original coins are modelled by $(\epsilon_n)$. When a guy encounters a coin he creates a new random variable $X_n = \left\{\begin{array}{cl} \epsilon'_n & \text{ if$\epsilon_n=1$}\\ 1 & \text{ if$\epsilon_n=-1$} \end{array} \right.$. Check that $(X_n)$ is a sequence of independent random variables.

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Assume that equilibrium has been reached with a fraction t of tails (and (1-t) of heads)

Do you know the probability of encountering a tail, and the probable change in tail population because of that encounter?

Ditto for encountering a head? (still interested in the change in tail population)

What overall change in tail population exists at equilibrium?

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thank-you for the answer, though I'm a little more confused. I've tried just assuming there is only 1 coin and that after N iterations what is the probability that the coin is a tail. that leave me with something as follows: (1/2)^N, which seems to suggest that probability of a tails is diminishing. – Hazen Jan 30 '13 at 5:24
could you please elaborate on your first comment relating the possible fraction of tails. – Hazen Jan 30 '13 at 5:24
Your question assumed that a long time had passed, and that some stable ratio of heads to tails had been reached. I may have confused things by using $0<t<1$ as the fraction of the whole collection that was showing tails at some particular instant. – DJohnM Jan 30 '13 at 5:33