TITLE: Order in a Biased Coin
QUESTION [6 upvotes]: Situation: 
Consider the classic coin tossing experiment. We want to explore if the coin is biased. 
Coin 1: Coin is tossed $50$ times. We get $20$T and then $30$H, in that sequence
Coin 2: Coin is tossed $50$ times. We get $4$T $6$H, $4$T $6$H, $4$T $6$H, $4$T $6$H, $4$T $6$H in that sequence. 
Q1 : Is the probability of Coin 1 and Coin 2 being biased is the same? My gut feel is that yes, because each event is independent so the order of the events doesn't matter at all. 
Q2:  I have a coin toss where observations are not independent. P(H | Previous toss is tail) = $0.6$ and P(T | Previous Head) = $0.5$  What kind of statistical test can I use to check the probability of coin being biased?

REPLY [2 votes]: A detailed answer has already been given under the assumption that the independence of the coin tosses is in question. I interpreted the question to mean that the independence assumption stands and you merely want to consider the possibility of the coin being biased away from the uniform distribution for $p=\frac12$.
In this case, the answer to Question $1$ is affirmative. In this experiment, the number of heads is a sufficient statistic for the parameter $p$ of interest, the probability of the coin to yield heads. That is, it contains all the information about this parameter that the entire run of data contains. Formally, the likelihood of the parameter conditional on the statistic is the same as the likelihood of the parameter conditional on the entire data. Thus, the order in which the results occur makes no difference in your posterior beliefs about the parameter.