TITLE: Confused about fair coins?
QUESTION [5 upvotes]: Bob and Alice play with a coin.
Together they have thrown $30$ heads and $70$ tails.
Bob says the coin is not fair.
Alice disagrees and proposes to keep tossing.
They toss another $100$ times and end Up with $100$ heads and $100$ tails.
Bob admits he is surprised but argues about the fairness anyway.
He says there are $2^{200}$ possible outcomes and $\binom {200}{100} $ of them are $100$ heads and $100$ tails.
So the probability that you are correct in claiming it has a probability of heads over tails between $\frac {99}{200}$ and $\frac {101}{200}$ is very close to $A = 1 - \frac{\binom {200}{100}}{2^{200}}$.
Cindy comes along , draws a Pascal triangle and points out that with a fair coin you have a probability of getting 100 heads equal to $A$.
Danny joins and claims that - because of what Cindy Said - it works in reverse too , if your probability is $A$ then your coin must be fair.
Emmet joins the discussion and says Danny is wrong , because he does not take into account the unfair coins.
Who is right , who is wrong ?
What is the truth ? 
What are the probabilities ?
How certain are we that the coin is fair ?

REPLY [2 votes]: (Edited. The first answer was wrong.)
The probability $P$ that a fair coin shows  $\leq30$ heads or tails in the first $100$ throws  is given by
$$P={2\over 2^{100}}\sum_{k=0}^{30}{100\choose k}\doteq0.0000785014\ ,$$
and the probability that this occurs two times in a row is $P^2\doteq6.16\cdot10^{-9}$.
Rejecting the null-hypothesis ("This coin is fair") after such an outcome means that you err in $0.000000616\%$ of the cases where actually a fair coin was presented to you.