# Is the coin fair

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?

• I think that any statistical test under the hypothesis that the coin is fair would lead to the conclusion that there is no ground to reject this hypothesis. Further I tend to say that nevertheless the probability of fairness of the coin is zero. Stronger: fair coins do not even exist. Jul 15 '19 at 8:17
• The professor isn't testing for fairness, he is testing for independence. The way this story is usually told is that half the students actually flip coins and the other half do their best to simply write down what looks like a random sequence. The professor tries to tell which is which by looking at the runs. Jul 15 '19 at 12:44

• @David Only if the runs of both sides are comparably long. Even an unfair coin with $P(H)<0.5$ can expect long runs of heads, if enough repetitions are done. Also, in probability, "fair" has an exact meaning. For coins, it means $P(H)=P(T)=\frac12$.
• @David Of course it is not. Because in a coin where tosses are not independent, $P(H)=\frac12$ is not a full and accurate description of that coin.