# 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?

• Each coin toss is independent of what came before it, however unlikely. – Ansari Apr 19 '13 at 1:05
• According to LaPlace's rule of succession, the conditional probability of getting a Head on the next toss, given that the first $100$ tosses resulted in Heads is $\frac{101}{102}$. The coin does not have to fair any more than the sun is. – Dilip Sarwate Apr 19 '13 at 2:04
• "The proportion of head to tails is 1 to 1 as number of trails approach infinity (this is a fact correct?)" - it's only a fact if the coin and tossing process is actually fair. You might make a symmetry argument in support of making this assumption, but it won't make a coin that's not quite fair become fair. The rest of your question is well covered already, so I won't labor that point. – Glen_b Apr 19 '13 at 2:42

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.

Let's discuss what this "force" exactly is. For this I find it easy to consider an actual (but fair) coin. Consider the following:

1. The coin has two stable states: heads and tails (let's ignore the fact it could land on its side).
2. 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".
3. 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).

• A "coin flip" is usually used as a metaphor for Binomial processes; as an object of study in its own right, it is at best of passing interest and a mere curiosity. Thus, any appeal to physical "forces" or other such explanations is unnecessarily limited. As such, I don't believe your answer really addresses the crux of the matter. – whuber Apr 19 '13 at 3:37
• @whuber of course there is no actual coin, but the OP specifically referred to some mystical "force". Therefore, while the mathematical explanation is straightforward and simple, it seemed to me that the OP might be misled by some physical intuition. I thought it would be helpful to specifically address the "force" intuition with a physical example. – Bitwise Apr 19 '13 at 3:45
• A fascinating paper about the physics of coin tossing: Diaconis et al, "Dynamical bias in the coin toss". – Scortchi Apr 19 '13 at 11:47
• @Bitwise I was using the coin flip as a metaphor for the Binomial processes, as whuber stated, so any process with p = 0.5 and !p = q = 0.5 – SGM1 Apr 19 '13 at 16:06
• Chaotic systems are hard to predict because they're extremely sensitive to initial conditions. But, they're actually completely deterministic. If you had perfect information and sufficient computational power, you could predict the entire future of the process from its initial conditions. I would argue that apparent randomness in this case doesn't come from true physical randomness, but from our lack of knowledge about the physical factors that determine the outcome. – user20160 Jun 5 '16 at 21:29