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Someone walks into your room and dumps a huge bag of quarters all over the floor. They spread them out so no quarters are on top of any other quarters. a robot then comes into the room and is programmed such that if it sees a head, it flips it to tails. If it sees a tail, it throws it in the air. the robot moves around randomly forever. Will there be a convergence in distribution of heads vs. tails?
Hmmm. I made a little math trick out of it. If the ‘bot finds a head, it flips. If the bot finds a tail, there’s a fifty percent chance this will become a head, as well.
(P_h = #heads/#coins, P_t = #tails/#coins)
So, delta h = -P_h + .5 P_t
= -(1 - P_t) + .5 P_t
= 1.5 P_t -1
Which is zero when P_t is 2/3. It’s important to remember that a flip to a tail results in no change to the number of tails — this threw me off for a second.