So, I told my friend a story …
Probability professor assigned a homework to his students. The assignment was to record a 200 tosses of the fair coin. After the assignments were handed, the professor split all of them in two heaps after briefly looking at each. One of the heaps was claimed to contain all the assignments of students inventing the experiment and not actually carrying it.
The explanation given [to us] is that a naturally carried experiment will end up with long sequences of $8$ (let say Heads) or more, which are totally missing if the results are being invented, as regular people suppose sequence of $8$ is way a too rare event.
So, I've decided to check this anecdote by myself. Those are my recorded results
THTHHHHHHHHHTTTTHHHTTHHTHTTTTTTTTHHTTTTHHTHTTHHHHTHTHTTTTHHTHHTHTHHTTHTHHHHHHTHHHTTHTHTTTHHTHTHTHHTTTHHHTTHHHTHHHTHTHHTHHTHTHTTTHTTTTHHHTTTTHHHHTTTTTTTHTHHHHHHTTHTHTHTHTTTHHHTTTTHHTHTHHTHHTTHTTHTTTHHH
There are $102$ Heads, with a longest run of $9$, and the longest run of Tails being $8$.
As I'm not a physicist, I suppose the experiment was not controlled, nor exactly reproducible. I've tossed coin high and low, starting with Heads up, then sometimes Tails, never invested a thought in how should I actually toss.
Now, my question is
Can we postulate, that the actual coin I've used is fair? Or what is the probability it is?
Best Answer
A lack of long runs of heads is "proof" (more like indication) that the string of heads and tails was not generated by using a series of independent coin tosses. It does not indicate that the coin was fair, and it does not indicate that it was not fair.
Similarly, the fact that long runs of heads exist is not proof that a coin is fair. In fact, unfair coins produce on average even longer runs of one of the two sides.