# 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. – drhab Jul 15 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. – awkward Jul 15 at 12:44

## 1 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.

• Right, but if by "unfair" we mean "more of one output, with tosses independent form each other", long runs of both heads and tails "indicates" fairness – David Jul 15 at 8:44
• @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 $P(H)=P(T)=\frac12$. – 5xum Jul 15 at 9:13
• You are right. After enough tosses of a tail-biased coin, you would eventually get very long runs of heads, but at that point you (probably) would've gotten even longer runs of tails. Also, are you sure that a coin where tosses are not independent is considered "fair"? – David Jul 15 at 9:26
• @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. – 5xum Jul 15 at 9:30