Statistics of a simple coin toss Original simple question:
"Given a fair coin that has been tossed 100 times, each time landing heads, would it be more likely that that the next coin flip be tails or heads"
The proportion of head to tails is 1 to 1 as number of trails approach infinity (this is a fact correct?). Therefore, if this must be the case, mustn't there be an enacting "force" per say that causes this state of being to (admitted unreachable, but technically eventual) case? Hence if there exist a force pushing the number of heads and tails towards one another, shouldn't there exist a higher chance of tails than (regardless of how significant) heads? Or is this thought process just illegitimate simple because the state exist on at the infinity case?
 A: What you describe is called the Gambler's fallacy and is a common misunderstanding.
The converging to 1 to 1 happens due to the 100 observations that you have already seen being overwhelmed, not compensated for.
A: Let's discuss what this "force" exactly is. For this I find it easy to consider an actual (but fair) coin. Consider the following:


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*The coin has two stable states: heads and tails (let's ignore the fact it could land on its side).

*In what sense is the outcome random when you throw a coin? If we threw the coin exactly in the same manner, we can agree it would fall on the same side each time. So why is it random? This is due to something known as "chaos": a very small change in the way you throw the coin (the initial condition of the system) can change the side it falls on. Essentially it makes the out come very difficult to predict, and we think of it as "random".

*Where do the probabilities come from? It turns out, that due to the physical structure of the coin, in half of the initial conditions you will get heads and in the other half you will get tails.


So now that we understand what really are the forces in play, we can answer your question. Each time you throw the coin, the same forces act but you slightly change the initial conditions. This is why each throw is independent, and even after 1000 heads your probability of getting a head the next throw is 0.5 (assuming the coin is fair, of course).
