Chris Lakomy, of Waikanae, asks :-
If I could repeatedly toss a coin in exactly the same way, would the coin always end up showing exactly the same face?
Richard Arnold, a statistician at Victoria University of Wellington, responded.
The answer to this is Yes – and No.
It’s Yes because the laws of physics are deterministic. If the conditions are the same, and you do the same thing, then the laws of physics say that the same outcome will always occur (at least for large objects like coins, much larger than an individual atom).
However for all practical purposes the answer is actually No. The reason is that you can never guarantee that you’ll toss the coin with exactly the same force, in exactly the same direction, with the air currents in the room moving in exactly the same way, and that you catch it exactly in the same way. And by exactly I do mean exactly.
They key idea here is that any differences, even very tiny ones, in the launching of the coin, and the air it passes through, can lead to a big difference to the final outcome.
This puts this question right at the boundary of physics and statistics, in the realm of a branch of mathematics known as Chaos Theory. No matter how carefully we specify the starting conditions of a Chaotic System, we can’t predict its future with certainty. Instead we can only talk about probabilities – in this case the 50% probability of a coin coming up heads.
Pendulums aren’t chaotic (which makes them good for keeping time), but coin tosses are. The weather is a chaotic system, which is the reason it’s impossible to predict the weather more than 10-14 days ahead.
Try building a mechanical coin tossing machine in a vacuum in a temperature controlled box – and you’ll still fail to replicate the same outcome on every coin toss. Among other things, the bearings in your machine will age and give the coin a slightly different launch velocity each time.