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:
- Move about and pick up a random coin.
- If the coin shows heads, gently turn it over and put it back down tails.
- If the coin shows tails, flip it high into the air, and let it randomly land (equal chance of landing heads or tails).
- 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?
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Solution
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:
- 1/3 of the coins (which were heads) will flip to tails.
- 1/2 × 2/3 of the coins will flip to tails (previously tails).
- 1/2 × 2/3 of the coins will flip to heads (previously tails).
(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.