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Coin Flipping Robot

There is a large warehouse. Strewn all over the warehouse floor are thousands of coins. The coins are unbiased, each with a heads side, and a tails side. Initially there is a random mix of heads and tails.
Into this warehouse is placed a robot. The robot has been programmed with a very simple set of instructions:
  1. Move about and pick up a random coin.
  2. If the coin shows heads, gently turn it over and put it back down tails.
  3. If the coin shows tails, flip it high into the air, and let it randomly land (equal chance of landing heads or tails).
  4. Goto 1
If we leave the robot to do his thing for a long, long, time, what is will happen to the ratio of tails to heads for the coins in the room?
After a long time, what will be the percentage of tails-up coins in the room?


It’s possible to solve this problem using a Markov Chain, but it’s such a simple silly little puzzle that you can solve it using just a little common sense and come critical thinking.
The answer is that the percentage of tails-up coins in the room will trend towards 2/3rds.
The room will reach equilibrium when 2/3rds of the coins are tails (meaning one third are heads). When it is in this state:
(We started with 2/3rds tails, and the expected outcome is still 2/3rds tails).
Another way to think about this is that the room will reach equilibrium when the chance of a tail turning into a head is the same as a head turning into a tail. If h is the proportion of heads, the chance of head to tail is just h. If t is the proportion of tails, then t = 1 - h. The chance of a tail turning to a head is ½(1 - h); It’s a coin flip! For these two to equal, h = ½(1 - h), so h = 1/3.